Geometry of Time-spaces O. A. Laudal

Geometry of Time-spaces



O. A. Laudal



REPORT No. 15, 2006/2007

ISSN 1103-467X

ISRN IML-R- -15-06/07- -SE

GEOMETRY OF TIME-SPACES.





Olav Arnfinn Laudal

Institute of Mathematics

U niv ersity of O slo



Abstract. In this paper we study the geometry of moduli spaces of representa-

tions of associativ e algeb ras A ov er a not necessarily algeb raically closed, fi eld k.

U sing the b asic concepts and the philosophy of classical physics we hav e, in [L a 6 -7 ],

constructed a non-commutativ e phase space A → P h (A) and its infi nite iteration

P h ∞ (A), with a canonical deriv ation δ, the D irac deriv ation. Introducing general

parsimony principles, in the form of L agrangians, or a force law, we ob tain, for ev ery

positiv e integer n, general eq uations of motions on the moduli space S im p n (A) of

n-dimensional simple modules ov er A , together with a general ev olution operator in

the space of sections of the corresponding, rank n, univ ersal b undle. T he law s o f

d y n am ics of S im p n (A), and in the space of sections of its univ ersal b undle, may b e

giv en the same form as in standard q uantum theory. T ime is, in accordance with the

philosophy of [L a 6 ], defi ned b y the D irac deriv ation, inducing a v ectorfi eld on the

moduli space S im p n (A).

M oreov er we show that the ex istence of a P lanck ’s constant is part of this picture.

S tudying, as an ex ample, the harmonic oscillator, in all rank s, a F ock representation,

and a generaliz ed notion of v ertex algeb ra pop up.

W e also include some simpler ex amples, lik e the one dimensional q uartic anhar-

monic oscillator in rank 2 and 3 .









§0 In tro d u c tio n .

In a fi rst p ap er on this sub ject, see [L a 6 ], w e sk etched a p hy sical ” toy mod el” ,

w here the sp ace-time of classical p hy sics b ecame a section of a univ ersal fi b er sp ace

˜

E, d efi ned on the mod uli sp ace, H, of the p hy sical sy stems w e chose to consid er,

in this case the sy stems comp osed of an ob serv er and an ob serv ed , b oth sitting in

E uclid ean 3 -sp ace. T his mod uli sp ace w as called the time-space.

Measurab le time, in this mathematical mod el, turned out to b e a metric ρ on the

time-sp ace, measuring all p ossib le infi nitesimal chang es of th e state of the ob jects

in the family w e are stud y ing .

T his lead to a ” p hy sics” w here there are no infi nite v elocities, and w here the

p rincip le of relativ ity comes for free. In p articular, w e ob serv e that the three

fund amental ” g aug e” g roup s of current q uantum theory U (1), S U (2 ) and S U (3 )

are p art of the structure of the fi b er sp ace,

˜

E− → H.

W ork partly done during the M oduli T heory-year 2 0 0 6 -0 7 at Institute M ittag-L effl er. M ath-

ematics S ub ject C lassifi cation (2 0 0 0 ): 1 4 A 2 2 , 1 4 H 5 0 , 1 4 R , 1 6 D 6 0 , 1 6 G 3 0 , 8 1 , 8 3 . K eywords:

A ssociativ e algeb ras, modules, simple modules, ex tensions, deformation theory, moduli spaces,

non-commutativ e algeb raic geometry, time, q uantum theory.



T ypeset b y AMS-T EX

1

2 OLAV ARNFINN LAUDAL



With this model in mind we embarked on the study of moduli spaces of mod-

ules ov er non-commutativ e alg ebras in g eneral. T he basic notions of the affi ne

non-commutativ e alg ebraic g eometry related to a (non-commutativ e) associativ e

k-alg ebra hav e been treated in sev eral tex ts, see [L a 2,3 ,4 ,5 ]. G iv en a fi nitely g en-

erated alg ebra A, we associate a non-commutativ e scheme-structure on the set of

isomorphism classes of simple fi nite dimensional representations, i.e. rig ht mod-

ules, Sim p → S im p n (A(σ))/ = : Xn .



T h e ore m (3 .2 ) and T h e ore m (3 .3 ), sh ou ld , in g ood c ase s, p rod u c e a rank n b u nd le

Vn on Yn , and th e re sh ou ld also e x ist a natu ral m orp h ism ,



Γn : Yn → M,



w h e re M is th e m od u li sp ac e of (alg e b raic ) c u rv e s. O ne m ig h t th e re fore try to fi nd

c ond itions for th e e x iste nc e of a rank n b u nd e l Nn , on M, for w h ich Γ∗ (Nn ) = Vn .

n

T h e se are q u e stions c lose ly re late d to th e notion of v e rte x alg e b ras. S e e th e v e ry

te ntativ e re m ark on th e se p rob le m s, at th e e nd of §3 .

P e rfe c tly p aralle l w ith th is th e ory of sim p le fi nite d im e nsional re p re se ntations, w e

m ig h t h av e c onsid e re d , for a g iv e n alg e b ra A, th e sp ac e of alg e b ra h om om orp h ism s,



κ:A→R .

GEOMETRY OF TIME-SPACES. 5



Any such homomorphism induces a unique commutative diagram of algebras,



κ /R

A



 P hκ



P h (A) / P h (R).



M oreover, since any derivation ξ ∈ D e r k (A, R) has a natural lifting to a deriva-

tion ξ ∈ D e r k (P h (A), P h (R)), by just defi ning ξ(da) = d(ξ(a)) w e fi nd, using

the general machinery of deformations of diagrams, see [L a 0 ], that any family of

morphisms κ induces a family of the above diagram.

A g-string might have been defi ned as any algebra R together w ith a pair of

P h-points, i.e. a pair of homomorphisms i : P h (R) → k(pi ), corresponding to tw o

points (pi ∈ S im p1 (R), i = 1 , 2, each outfi tted w ith a tangent ξi . W e might have

considered any tw o points pi ∈ S im pn (P h (R)), but let us just consider the case

n = 1 . F or any g-string, consider the no n-co m m u ta tiv e ta nge nt sp a ce of the the

pair of points,

T (R, p1 , p2 ) := E x t1 h (R) (k(p1 ), k(p2 )).

P



It is reasonable to call it the sp a ce o f te nsio ns betw een the tw o points of the string.

U sing this, w e include at the end of §2, a remark on generaliz ed strings, the

moduli space of such, including proposed von N eumann and D irichlet conditions.

In §3 , w e prove the main results, and w e add some simple ex amples, and fi nally

in §4 , w e shall introduce notions lik e inte ra c tio ns, life tim e , d eca y and c rea tio n o f

p a rtic le s.

W hen A is the coordinate k-algebra of a moduli space, w e should also consider

the family of L ie algebras of e sse ntia l automorphisms of the objects classifi ed by

S pe c(A), and obtain a general form for Y ang-M ills theory, see [B j-L a] and [L a-P f],

for the case of plane curve singularities. T his off ers us a model for the notion of

ga u ge p a rtic le s a nd ga u ge fi e ld s, coupling w ith ordinary particles via representations

onto the corresponding simple modules. In a (hopefully) forthcoming paper, w e

shall go back to our ” toy” model, w here the standard G auge groups, U (1 ), S U (2),

and S U (3 ) pop up canonically, and show that the results above can be used to

construct a general geometric theory closely related to general relativity and to

quantum theory, generaliz ing both.

§1 Phase spaces and the Dirac derivation.

G iven a k-algebra A, denote by A/ k − alg the category w here the objects are

homomorphisms of k-algebras κ : A → R, and the morphisms, ψ : κ → κ are

commutative diagrams,

AA

 AA

κ  κ

AA

 AA



R /R

ψ



and consider the functor,



D e r k (A, −) : A/ k − alg −→ S e ts .

6 OLAV ARNFINN LAUDAL



It is representable by a k-algebra-morphism,



ι : A −→ P h (A),



w ith a universal family giv en by a u niv ersal d eriv ation,



d : A −→ P h (A).



P h (A ) is relativ ely easy to c ompu te. It c an be c onstru c ted as the non-c ommu tativ e

v ersal base of the d eformation fu nc tor of the morphisme ρ : A → k[ ], see [L a 6 ]

and [L a 7 ].

C learly w e hav e the id entities,



d∗ : D e r k (A, A) = M o r A (P h (A), A),



and ,

d∗ : D e r k (A, P h (A)) = E n dA (P h (A)),

the last one assoc iating d to the id entity end omorphisme of P h . L et now V be a

right A-mod u le, w ith stru c tu re morphism ρ : A → E n dk (V ). W e obtain a u niv ersal

d eriv ation,

c : A −→ Ho m k (V, V ⊗A P h (A)),

d efi ned by, c(a)(v) = v ⊗ d(a). U sing the long ex ac t seq u enc e, see the introd u c tion,



0 → Ho m A (V, V ⊗A P h (A)) → Ho m k (V, V ⊗A P h (A)) →ι

D e r k (A, Ho m A (V, V ⊗A P h (A))) →κ E x t1 (V, V ⊗A P h (A)) → 0,

A





w e obtain the non-c ommu tativ e K od aira-S penc er c lass,



c(V ) := κ(c) ∈ E x t1 (V, V ⊗A P h (A)),

A





ind u c ing the K od aira-S penc er morphism,



g:Θ A := D e r k (A, A) −→ E x t1 (V, V ),

A



˜

v ia the id entity, δ∗ . If c(V ) = 0, then the ex ac t seq u enc e abov e prov es that there

ex ist a ˜

∈ Ho m k (V, V ⊗A P h (A)) su ch that δ = ι( ). T his is ju st another w ay

of prov ing that δ ˜ is giv en by a c onnec tion,



: D e r k (A, A) −→ Ho m k (V, V ).



A s is w ell know n, in the c ommu tativ e c ase, the K od aira-S penc er c lass giv es rise to

a C hern charac ter by pu tting,



ch i (V ) := 1 / i! ci (V ) ∈ E x ti (V, V ⊗A P h (A)),

A





and if c(V ) = 0, the c u rv atu re R(V ) ind u c es a c u rv atu re c lass,



R ∈ H 2 (k, A; Θ A, E n dA (V )).

GEOMETRY OF TIME-SPACES. 7



Any P h(A)-m o d u le W , g iv e n b y its stru c tu re m a p ,



ρW : P h(A) −→ E n dk (W )



c o rre sp o nd s b ije c tiv e ly to a n ind u c e d A-m o d u le stru c tu re o n W , a nd a d e riv a tio n

δρ ∈ D erk (A, E n dk (W )), d e fi ning a n e le m e nt,



[δρ ] ∈ E xt1 (W, W ),

A





se e th e intro d u c tio n. F ix ing th is e le m e nt w e fi nd th a t th e se t o f P h(A)-m o d u le

stru c tu re s o n th e A-m o d u le W is in o ne to o ne c o rre sp o nd e nc e w ith ,



E n dk (W )/E n dA (W ).



C o nv e rse ly, sta rting w ith a n A-m o d u le V a nd a n e le m e nt δ ∈ D erk (A, E n dk (V )),

w e o b ta in a P h(A)-m o d u le Vδ . It is th e n e a sy to se e th a t th e k e rne l o f th e na tu ra l

m ap ,

E xt1 h (A) (Vδ , Vδ ) → E xt1 (V, V ),

P A



ind u c e d b y th e line a r m a p ,



D erk (P h(A), E n dk (Vδ )) → D erk (A, E n dk (V ))



is th e q u o tie nt,

D erA (P h(A), E n dk (Vδ ))/E n dk (V ).

Remark. S inc e E xt1 (V, V ) is th e ta ng e nt sp a c e o f th e m iniv e rsa l d e fo rm a tio n sp a c e

A

o f V a s a n A-m o d u le , w e se e th a t th e no n-c o m m u ta tiv e sp a c e P hA a lso p a ra m e triz e s

th e se t o f g en eralized mo men ta, i.e . th e se t o f p a irs o f a sim p le m o d u le V ∈

S im p (A), a nd a ta ng e nt v e c to r o f S im p (A) a t th a t p o int.

Example 1.1. (i) L e t A = k[t], th e n o b v io u sly, P h(A) = k a nd d is

g iv e n b y d(t) = dt, su ch th a t fo r f ∈ k[t], w e fi nd d(f ) = Jt (f ) w ith th e no ta tio ns

o f [L a 5 ], i.e . th e no n-c o m m u ta tiv e d e riv a tio n o f f w ith re sp e c t to t. O ne sh o u ld

a lso c o m p a re th is w ith th e no n-c o m m u ta tiv e T a ylo r fo rm u la o f lo c .c it. If V k2

is a n A-m o d u le , d e fi ne d b y th e m a trix X ∈ M2 (k), a nd δ ∈ D erk (A, E n dk (V )),

is d e fi ne d in te rm s o f th e m a trix Y ∈ M2 (k), th e n th e P h(A)-m o d u le Vδ is th e

k -m o d u le d e fi ne d b y th e a c tio n o f th e tw o m a tric e s X, Y ∈ M2 (k), a nd

w e fi nd



e1 : = dim

V kE xt1 (V, V ) = dim

A kE n dA (V ) = dim k {Z ∈ M2 (k)| [X, Z] = 0 }

e1 δ

V : = dim kE xt1 h (A) (Vδ , Vδ )

P = 8 − 4 + dim {Z ∈ M2 (k)| [X, Z] = [Y, Z] = 0 }.



W e h a v e th e fo llo w ing ine q u a litie s,



2 ≤ e1 ≤ 4 ≤ e1 δ ≤ 8.

V V





(ii) L e t A = k 2 k[x]/(x2 − r2 ), r ∈ k, r = 0 , th e n,



P h(A) = k /((x2 − r2 ), x · dx + dx · x).

8 OLAV ARNFINN LAUDAL



Notice that P h(A) ju st has 2 p oin ts, i.e. sim p le rep resen tation s, g iv en b y ,



k(r) : x = r, dx = 0, k(−r) : x = −r, dx = 0.



A n easy com p u tation show s that,



E xt1 h(A) (k(α), k(α)) = 0, α = r, −r, E xt1 h(A) (k(α), k(−α)) = k · ω,

P P





w here ω is rep resen ted b y the d eriv ation g iv en b y ω(x) = 2 r, ω(dx) = t ∈ k w here

t is the tension of this string of dimension −1, see end of §2 , a nd end of §3 . N otice

a lso tha t this is a n ex a mp le of the ex istence of ta ngents betw een diff erent p oints, in

non-commu ta tiv e a lgeb ra ic geometry .

(iii) N ow , let A = k[x1 , x2 , x3 ] a nd consider,



P h(A) = k / ([xi , xj ], d([xi , xj ])),



a nd consider a ny 2 -dimensiona l rep resenta tion of P h(A). It is a n ea sy comp u ta tion

tha t a ny su c h is giv en b y the a c tions,



a1 0 b1 0 c1 0

x1 = , x2 = , x3 = ,

0 a2 0 b2 0 c2



a nd,



α1,1 (a1 − a2 )

dx1 = ,

(a2 − a1 ) α2,2

β1,1 (b1 − b2 )

dx2 = ,

(b2 − b1 ) β2,2

γ1,1 (c1 − c2 )

dx3 =

(c2 − c1 ) γ2,2



T he a ngu la r momentu m is now giv en b y ,



(a1 β1,1 − b1 α1,1 ) (a2 b1 − a1 b2 )

L1,2 := x1 dx2 − x2 dx1 = ,

(a1 b2 − a2 b1 ) (a2 β2,2 − b2 α2,2 )



etc . A nd the isosp in ha s the form,



0 (a1 − a2 )2

I1 := [x1 , dx1 ] = ,

(a2 − a1 )2 0



etc .

T he p ha se-sp a ce constru c tion ma y , of cou rse, be itera ted. G iv en the k-a lgeb ra A

w e ma y form the seq u ence, {P hn (A)}1≤n , defi ned indu c tiv ely b y



P h0 (A) = A, P h1 (A) = P h(A), ..., P hn+ 1 (A) := P h(P hn (A)).



L et in : P hn (A) → P hn+ 1 (A) be the ca nonica l imbedding, a nd let dn : P hn (A) →

0

P hn+ 1 (A) be the corresp onding deriv a tion. S ince the comp osition of in a nd the

0

GEOMETRY OF TIME-SPACES. 9



derivation dn+1 is a derivation P h n (A) → P h n+2 , th ere ex ist b y u niversality a

h om om orp h ism in+1 : P h n+1 (A) → P h n+2 (A), su c h th at,

1





dn ◦ in+1 = in ◦ dn+1 .

1 0





N otice th at w e h ere com p ose fu nc tions and fu nc tors from left to rig h t. C learly w e

m ay continu e th is p rocess constru c ting new h om om orp h ism s,



{in : P h n (A) → P h n+1 (A)}0≤j≤n ,

j





w ith th e p rop erty ,

dn ◦ in+1 = in ◦ dn+1 .

j+1 j



It is easy to see, [L a 7 ], th at,



in in+1 = in in+1 , p ), is a sch e m e fo r k .

be th e free k-a lg eb ra o n d sy m bo ls, a n d

let V ∈ S im p n (A). T h en



H A (V )c o m H A(n) (V ) k[[t1 , ..., t(d−1)n2 + 1 ]]



T h is sh o u ld b e c o m p a re d w ith th e re su lts o f [P ro c e si 1 ], se e a lso [F o rm a n e k ].

In g e n e ra l, th e n a tu ra l m o rp h ism ,



η(n) : A(n) → H A(n) (V ) ⊗k E ndk (V )

V ∈S im p n (A)





is n o t a n in je c tio n .

E x a m p le 2 .1 2 . In fa c t, le t

 

k k k

A = k k k.

0 0 k



T h e id e a l I(2 ) is g e n e ra te d b y [e1,1 , e1,2 , e2,2 , e2,3 ] = e1,3 . S o

   

k k k 0 0 k

A(2 ) =  k k k/0 0 k M2 (k) ⊕ M1 (k).

0 0 k 0 0 0



H o w e v e r,

H A(2) (V ) ⊗k E ndk (V ) M2 (k),

V ∈S im p 2 (A)



th e re fo re ker η(2 ) = M1 (k) = k.

L e t O(n), b e th e im a g e o f η(n), th e n ,



O(n) ⊆ H A(n) (V ) ⊗k E ndk (V )

V ∈S im p n (A)





a n d fo r e v e ry V ∈ S im p n (A),



H O(n) (V ) H A(n) (V ).



P ut B = V ∈S im p n (A) H A(n) (V ). C h o o sin g b a se s in a ll V ∈ S im p n (A), th e n



H A(n) (V ) ⊗k E ndk (V ) Mn (B),

V ∈S im p n (A)





L e t xi ∈ A, i = 1 , ..., d b e g e n e ra to rs o f A, a n d c o n sid e r th e ir im a g e s (xi ,q ) ∈

p

Mn (B). N o w , B is c o m m u ta tiv e , so th e k-su b -a lg e b ra C(n) ⊂ B g e n e ra te d b y th e

e le m e n ts {xi ,q }i= 1,..,d; p ,q = 1,..,n is c o m m u ta tiv e . W e h a v e a n in je c tio n ,

p





O(n) → Mn (C(n)),

GEOMETRY OF TIME-SPACES. 19



and for all V ∈ S im p n (A), w ith a ch ose n b asis, th e re is a natu ral c om p osition of

h om om orp h ism s of k-alg e b ras,



α : Mn (C(n)) → Mn (H A(n) (V )) → E nd k (V ),



indu c ing a c orre sp onding c om p osition of h om om orp h ism s of th e c e nte rs,



Z(α) : C(n) → H A(n) (V ) → k



T h is se ts u p a se t th e ore tic al inje c tiv e m ap ,



t : S im p n (A) − → M a x(C(n)),



de fi ne d b y t(V ) := ke r Z(α).

S inc e A(n) → H A(n) (V ) ⊗k E nd k (V ) is top olog ic ally su rje c tiv e , H A(n) (V ) ⊗k

E nd k (V ) is top olog ic ally g e ne rate d b y th e im ag e s of xi , i = 1 , ..., d . It follow s th at

w e h av e a su rje c tiv e h om om orp h ism ,



ˆ

C(n)t(V ) → H A(n) (V ).



C ate g oric al p rop e rtie s im p lie s, th at th e re is anoth e r natu ral m orp h ism ,



ˆ

H A(n) (V ) → C(n)t(V ) ,



w h ich c om p ose d w ith th e form e r is an au tom orp h ism of H A(n) (V ). S inc e



Mn (C(n)) ⊆ H A(n) (V ) ⊗k E nd k (V ),

V ∈S im p n (A)





it follow s th at for mv ∈ M a x(C(n)), c orre sp onding to V ∈ S im p n (A), th e fi nite

dim e nsional k-alg e b ra Mn (C(n)/mv 2 ) sits in a fi nite dim e nsional q u otie nt of som e ,



H A(n) (V ) ⊗k E nd k (V ).

V ∈V



w h e re V ⊂ S im p n (A) is fi nite . H ow e v e r, b y L e m m a (2 .5 ), th e m orp h ism ,



A(n) − → H A(n) (V ) ⊗k E nd k (V )

V ∈V



is top olog ic ally su rje c tiv . T h e re fore th e m orp h ism ,



A(n) − → Mn (C(n)/mv 2 )



is su rje c tiv , im p ly ing th at th e m ap



ˆ

H A(n) (V ) → C(n)mv ,



ˆ

is su rje c tiv , and c onse q u e ntly , H A(n) (V ) C(n)mv .

W e now h av e th e follow ing th e ore m , se e C h ap te r V III, §2 , of th e b ook [P roc e si

2 ], w h e re p art of th is th e ore m is p rov e d.

20 OLAV ARNFINN LAUDAL



Theorem 2.13. Let V ∈ S imp n (A), co rresp o n d to th e p o in t mv ∈ S imp 1 (C(n)).

(i) T h ere ex ist a Z a risk i n eig h bo rh o o d Uv o f v in S imp 1 (C(n)) su c h th a t a n y

c lo sed p o in t mv ∈ Uv co rresp o n d s to a u n iq u e p o in t V ∈ S imp n (A).

Let U (n) be th e o p en su b set o f S imp 1 (C(n)), th e u n io n o f a ll Uv fo r V ∈

S imp n (A).

(ii) O(n) d efi n es a n o n -co m m u ta tiv e stru c tu re sh ea f O(n) := OU (n) o f A zu m a y a

a lg eb ra s o n th e to p o lo g ica l sp a ce U (n) (J a co b so n to p o lo g y ).

(iii) T h e cen ter S(n) o f O(n), d efi n es a sc h em e stru c tu re o n S imp n (A).

˜

(iv ) T h e v ersa l fa m ily o f n-d im en sio n a l sim p le m o d u les, V := C(n)) ⊗k V , re-

stric ted to U (n), is d efi n ed b y th e m o rp h ism ,



˜

ρ : A → O(n) ⊆ E nd C(n) (C(n)) ⊗k V ) Mn (C(n)).



˜

(v ) T h e tra ce rin g T r ρ ⊆ S(n) ⊆ C(n) d efi n es a co m m u ta tiv e q u a si-a ffi n e sc h em e

stru c tu re o n S imp n (A). M o reo v er, th ere is a m o rp h ism o f sc h em es,



κ : U (n) −→ S imp n (A),



su c h th a t fo r a n y v ∈ U (n),



H A(n) (V ) ˆ

S(n)κ(v) (T ˆ ρ)κ(v)

r˜ ˆ

C(n)v



P ro o f. L e t ρ : A −→ E nd k (V ) b e th e su rje c tiv e h o m o m o rp h ism o f k-alg e b ras,

d e fi n in g V ∈ S imp n (A). L e t, as ab o v e ei,j ∈ E nd k (V ) b e th e e le m e n tary m atri-

c e s, an d p ick yi,j ∈ A su ch th at ρ(yi,j ) = ei,j . L e t u s d e n o te b y σ th e c y c lic al

p e rm u tatio n o f th e in te g e rs {1, 2, ..., n}, an d p u t,



sk := [yσk (1),σk (2 ) , yσk (2 ),σk (2 ) , yσk (2 ),σk (3 ) , ..., yσk (n),σk (n) ], s := sk ∈ A.

k= 0 ,1,..,n−1





C le arly s ∈ I(n − 1). S in c e [eσk (1),σk (2 ) , eσk (2 ),σk (2 ) , eσk (2 ),σk (3 ) ...eσk (n),σk (n) ] =

eσk (1),σk (n) ∈ E nd k (V ), ρ(s) := k= 0 ,1,..,n−1 ρ(sk ) ∈ E nd k (V ) is th e m atrix w ith

n o n -z e ro e le m e n ts, e q u al to 1, o n ly in th e (σ k (1), σ k (n)) p o sitio n , so th e d e te rm in an t

o f ρ(s) m u st b e + 1 o r -1. T h e d e te rm in an t d et(s) ∈ C(n) is th e re fo re n o n z e ro at th e

p o in t v ∈ S p ec(C(n)) c o rre sp o n d in g to V . P u t Uv = D(d et(s)) ⊂ S p ec(C(n)), an d

c o n sid e r th e lo c aliz atio n O(n){s} ⊆ Mn (C(n){d e t(s)} ), th e in c lu sio n fo llo w in g fro m

g e n e ral p ro p e rtie s o f th e lo c aliz atio n . N o w , an y c lo se d p o in t v ∈ Uv c o rre sp o n d s to

a n-d im e n sio n al re p re se n tatio n o f A, fo r w h ich th e e le m e n t s ∈ I(n−1) is in v e rtib le .

B u t th e n th is re p re se n tatio n c an n o t h av e a m n . In

fac t, fo r an y fi n ite d im e n sio n al O(n){s} -m o d u le V , o f d im e n sio n m, th e im ag e s o f ˆ

s in E nd k (V ) m u st b e in v e rtib le . H o w e v e r, th e in v e rse s−1 m u st b e th e im ag e o f

ˆ

a p o ly n o m ial (o f d e g re e m − 1) in s. T h e re fo re , if V is sim p le o v e r O(n){s} , i.e . if

th e h o m o m o rp h ism O(n){s} → E nd k (V ) is su rje c tiv e , V m u st also b e sim p le o v e r

A. S in c e n o w s ∈ I(n − 1), it fo llo w s th at m ≥ n. If m > n, w e m ay c o n stru c t,

in th e sam e w ay as ab o v e an e le m e n t in I(n) m ap p in g in to a n o n z e ro e le m e n t o f

E nd k (V ). S in c e , b y c o n stru c tio n , I(n) = 0 in A(n), an d th e re fo re also in O(n){s} ,

GEOMETRY OF TIME-SPACES. 21



we have proved what we wanted. By a theorem of M. Artin, see [Artin], O(n){s}

mu st b e an Az u maya alg eb ra with c enter, S(n){s} := Z(O(n){s} ). T herefore O(n)

defi nes a presheaf O(n) on U (n), of Az u maya alg eb ras with c enter S(n) := Z(O(n)).

C learly, any V ∈ Sim p n (A), c orresponding to mv ∈ M ax(C(n)) maps to a point

κ(v) ∈ Sim p (O(n)). L et mκ(v) b e the c orresponding max imal ideal of S(n). S inc e

O(n) is loc ally Az u maya, it follows that,



ˆ

S(n)mκ(v) H O(n) (V ) H A(n) (V ).



T he rest is c lear.





Example 2.14. L et u s check the c ase of A = k , the free non-

c ommu tative k-alg eb ra on two symb ols. F irst, we shall c ompu te E xt1 (V, V ) for

A

a partic u lar V ∈ Sim p 2 (A), and fi nd a b asis {t∗ }5 1 , represented b y derivations

i i=

∂i := ∂i (V ) ∈ D er k (A, E nd k (V )), i=1,2,3,4,5. T his is easy, sinc e for any two

A-modu les V1 , V2 , we have the ex ac t seq u enc e,



0 → Ho m A (V1 , V2 ) → Ho m k (V1 , V2 ) → D er k (A, Ho m k (V1 , V2 ))



→E xt1 (V1 , V2 )

A →0



proving that, E xt1 (V1 , V2 ) = D er k (A, Ho m k (V1 , V2 ))/ T r iv, where T r iv is the su b -

A

vec tor spac e of trivial derivations. P ick V ∈ Sim p 2 (A) defi ned b y the homomor-

phism A → M2 (k) mapping the g enerators x1 , x2 to the matric es



0 1 0 0

X1 := =: e1,2 , X2 := =: e2,1 .

0 0 1 0



N otic e that



1 0 0 0

X1 X2 = =: e1,1 = e1 , X2 X1 = =: e2,2 = e2 ,

0 0 0 1



and rec all also that for any matrix (ap ,q ) ∈ M2 (k), ei (ap ,q )ej = ai,j ei,j . T he trivial

derivations are g enerated b y the derivations {δp ,q }p ,q = 1.2 , defi ned b y,



δp ,q (xi ) = Xi ep ,q − ep ,q Xi .



C learly δ1,1 + δ2,2 = 0. N ow, c ompu te and show that the derivations ∂i , i =

1, 2, 3, 4, 5, defi ned b y,



∂i (x1 ) = 0, for i = 1, 2, ∂i (x2 ) = 0, for i = 4, 5,



b y,



∂1 (x2 ) = e1,1 , ∂2 (x2 ) = e1,2 , ∂3 (x1 ) = e1,2 , ∂4 (x1 ) = e2,2 , ∂5 (x1 ) = e2,1



and b y,

∂3 (x2 ) = e2,1 ,

22 OLAV ARNFINN LAUDAL



form a basis for Ext1 (V, V ) = D er k (A, End k (V ))/ T r iv . S in c e Ext2 (V, V ) = 0 w e

A A

fi n d H(V ) = k > an d so H(V )c o m k[[t1 , t2 , t3 , t4 , t5 ]]. T h e

˜

formal v e rsal family V , is d e fi n e d by th e ac tion s of x1 , x2 , g iv e n by ,



0 1 + t3 t1 t2

X1 := , X2 := .

t5 t4 1 + t3 0



O n e ch e ck s th at th e re are p oly n omials of X1 , X2 w h ich are e q u al to ti ep ,q , mod u lo

ˆ

th e id e al (t1 , .., t5 )2 ⊂ H(V ), for all i, p , q = 1 , 2 . T h is p rov e s th at C(2 )v mu st be

isomorp h ic to H(V ), an d th at th e c omp osition ,



A −→ A(2 ) −→ M2 (C(2 )) ⊂ M2 (H(V )))



is top olog ic ally su rje c tiv e . B y th e c on stru c tion of C(n) th is also p rov e s th at



C(2 ) k[t1 , t2 , t3 , t4 , t5 ].



loc ally in a Z arisk i n e ig h borh ood of th e orig in . M ore ov e r, th e F orman e k c e n te r, in

th is c ase is c u t ou t by th e sin g le e q u ation :



f := d et[X1 , X2 ] = −((1 + t3 )2 − t2 t5 )2 + (t1 (1 + t3 ) + t2 t4 )(t4 (1 + t3 ) + t1 t5 ).



C omp u tin g , w e also fi n d th e follow in g formu las,



tr X1 = t4 , tr X2 = t1 ,

d etX1 = −t5 − t3 t5 , d etX2 = −t2 − t2 t3 ,

tr (X1 X2 ) = (1 + t3 )2 + t2 t5



so th e trace ring of th is family is



k[t1 , t2 + t2 t3 , 1 + 2 t3 + t2 + t2 t5 , t4 , t5 + t3 t5 ] =: k[u1 , u2 , ..., u5 ],

3



w ith ,



u1 = t1 , u2 = (1 + t3 )t2 , u3 = (1 + t3 )2 + t2 t5 , u4 = t4 , u5 = (1 + t3 )t5 ,



an d f = −u2 + 4u2 u5 + u1 u3 u4 + u2 u5 + u2 u2 . C(2 ) = k[t] is alg e braic ov e r k[u],

3 1 4

w ith d isc rimin an t, ∆ := 4u2 u5 (u2 − 4u2 u5 ) = 4(1 + t3 )2 t2 t5 ((1 + t3 )2 − t2 t5 )2 . F rom

3

e

th is follow s th at th e re is an ´ tale c ov e rin g



A5 − V (f ∆ ) → S im p 2 (A) − V (∆ ).



N otic e th at if w e p u t t1 = t4 = 0, th e n f = ∆ . S e e E x amp le (3 .7 )

C o m p letio ns o f S im p n (A). In th e e x amp le abov e it is e asy to se e th at e le me n ts

of th e c omp le me n t of U (n) in th e affi n e su b-sch e me S p ec(C(n)) may be re p re se n te d

e ith e r by in d e c omp osable , or d e c omp osable re p re se n tation s. A d e c omp osable re p -

re se n tation W w ill, h ow e v e r, n ot in g e n e ral be d e formable in to a simp le re p re se n -

tation , sin c e g ood d e formation s mu st c on se rv e End A (W ). T h e re fore , e v e n th ou g h

w e h av e te rme d S p ec(C(n)) a c omp ac tifi c ation of U (n), it is a bad co m p letio n. T h e

GEOMETRY OF TIME-SPACES. 23



missing points at infinity of U (n) or S im p n (A), sh ou ld b e re pre se nte d b y ind e c om-

posa b le re pre se nta tions, w ith E nd A (W ) = k. A ny su ch is a n ite ra te d e x te nsion of

simple re pre se nta tions {Vi }i=1,2,..s , w ith re pre se nta tion gra ph Γ (c orre spond ing to

s

a n e x te nsio n typ e , se e [L a 4 ]), a nd i=1 d im (Vi ) = n. T o simplify th e nota tions w e

o

sh a ll w rite , |Γ| := {Vi }i=1,2,..s . In [L a 2 ,4 ], se e a lso [J ¨ -L a -S l], w e tre a t th e prob le m

of c la ssify ing a ll su ch ind e c omposa b le re pre se nta tions, u p to isomorph isms. L e t u s

re c a ll th e ma in id e a s.

A ssu me th a t th e simple mod u le s {Vi } w e sh a ll ta lk a b ou t a re su ch th a t a ll

E x t1 (Vi , Vj ) a re fi nite d ime nsiona l a s k-v e c tor spa c e s. L e t Γ b e a n ord e re d gra ph

A

w ith se t of nod e s |Γ| = {Vi }. S ta rting w ith th e th e fi rst nod e of Γ, w e c a n c onstru c t,

in ma ny w a y s, a n e x te nsion of th e c orre spond ing mod u le Vi1 w ith th e mod u le Vi2

c orre spond ing to th e e nd point of th e fi rst a rrow of Γ, th e n c ontinu e , ch oosing a n

e x te nsion of th e re su lt w ith th e mod u le c orre spond ing to th e e nd point of th e se c ond

a rrow of Γ, e tc . u ntill w e h a v e re a ch e d th e e nd point of th e la st a rrow . A ny fi nite

le ngth mod u le c a n b e ma d e in th is w a y , th e ” opposite ly ord e re d ” Γ c orre spond ing

to a d e c omposition of th e mod u le into simple c onstitu e nc ie s, b y pe e ling off one

simple su b -mod u le a t a time , i.e . b y pick ing one simple su b -mod u le a nd forming

th e q u otie nt, pick ing a se c ond simple su b -mod u le of th e q u otie nt a nd ta k ing th e

q u otie nt, a nd re pe a ting th e proc e d u re u ntill it stops.

T h e ” ord e re d ” k-a lge b ra k[Γ] of th e ord e re d gra ph Γ is th e q u otie nt a lge b ra of

th e u su a l a lge b ra of th e gra ph Γ b y th e id e a l ge ne ra te d b y a ll a d missib le w ord s

w h ich a re not ” inte rv a ls” of th e ord e re d gra ph . S a y ...γi,j (n − 1 )γj,j (n)γj,k (n + 1 )...

is is a n inte rv a l of th e ord e re d gra ph , th e n γi,j (n − 1 ).γj,k (n + 1 ) = 0 in k[Γ].

N ow , le t H(|Γ|) b e th e forma l mod u li of th e fa mily |Γ|. W e sh ow in [L a 4 ], se e

P roposition 2 . a b ov e , th a t a ny ite ra te d e x te nsion of th e {Vi }r w ith e x te nsio n

i=1

typ e , i.e . gra ph , Γ c orre spond s to a morph ism in ar ,



α : H −→ k[Γ].



M ore ov e r th e se t of isomorph ism c la sse s of su ch mod u le s is pa ra me triz e d b y a

q u otie nt spa c e of th e a ffi ne sch e me ,



A(Γ) := M o r ar (H(|Γ|), k[Γ]).



L e t α ∈ A(Γ), a nd le t V (α) d e note th e c orre spond ing ite ra te d e x te nsion mod u le ,

th e n th e ta nge nt spa c e of A(Γ) a t α is,



TA(Γ ),α := D e r k (H(|Γ|), k[Γ]α ),



w h e re k[Γ]α is k[Γ] c onsid e re d a s a H(|Γ|)-b imod u le v ia α. T h e ob stru c tion spa c e

for th e d e forma tion fu nc tor of α is HH 2 (H(|Γ|), k[Γ]), a nd w e ma y , a s is e x pla ine d

in [L a 0 ,1 ], c ompu te th e c omple te loc a l ring of A(Γ) a t α. In pa rtic u la r w e ma y

d e c id e w h e th e r th e point is a smooth point of A(Γ), or not.

T h e a u tomorph ism grou p G of k[Γ], c onsid e re d a s a n ob je c t of ar , h a s a L ie

a lge b ra w h ich w e sh a ll c a ll g. O b v iou sly w e h a v e ,



g = D e r k (k[Γ], k[Γ]).



C le a rly a n ite ra te d e x te nsion α w ith gra ph Γ w ill b e isomorph ic a s A-mod u le to

g(α), for a ny g ∈ G. In pa rtic u la r, if δ ∈ g, th e n e x p (δ)(α) is isomorph ic to α a s

a n ite ra te d e x te nsion of A-mod u le s, w ith th e sa me gra ph a s α.

24 OLAV ARNFINN LAUDAL



Consider the map,



α∗ : De r k (k[Γ ], k[Γ ]) → De r k (H(Γ ), k[Γ ]α ).



T he imag e of α∗ is the su b spac e of the tang ent spac e of A(Γ ) at α along w hich the

c orresponding modu le has c onstant isomorphism c lass.

N otic e that if α is a smooth point, and α∗ is not su rjec tiv then there is a positiv e-

dimensional modu li spac e of iterated ex tension modu les w ith g raph Γ throu g h α.

Clearly the k ernel of α∗ is c ontained in the L ie alg eb ra of au tomorphisms of the

modu le V (α), and shou ld b e c ontained in E nd A (V (α)). F rom this follow s that if

V (α) is indec omposab le then ke r α∗ = 0 . T he E u ler ty pe deriv ations, defi ned b y ,



δE (γi,j ) = ρi,j γi,j , ρi,j ∈ k



are the easiest to check ! N otic e how ev er, that there may b e disc rete au tomorphisms

in G, not of ex ponential ty pe, leav ing α inv ariant. N otic e also that an indec ompos-

ab le modu le may hav e an endomorphism-ring w hich is a non-triv ial loc al ring .

A ssu me now that w e hav e identifi ed the non-commutative scheme of indec om-

posab le Γ -representation, c all it Ind Γ (A). P u t S im p Γ (A) := S im p n (A) ∪ Ind Γ (A).

N ow , repeat the b asic s of the c onstru c tion of S p e c(C(n)) ab ov e. Consider for ev ery

open affi ne su b scheme D(s) ⊂ S im p Γ (A), the natu ral morphism,



A → lim O(c, π )

← −

c⊂D(s)





c ru nning throu g h all fi nite su b sets of D(s). P u t Bs (Γ ) := V ∈D(s) H A(n) (V )co m ,

and c onsider the homomorphism,



A → A(n) → H A(n) (V )co m ⊗k E nd k (V ) Mn (Bs (Γ )).

V ∈D(s)





L et xi ∈ A, i = 1 , ..., d b e g enerators of A, and c onsider the imag es (xi ,q ) ∈ Bs (n)⊗k

p

E nd k (k n ) of xi v ia the homomorphism of k-alg eb ras,



A → Bs (Γ ) ⊗ Mn (k),



ob tained b y choosing b ases in all V ∈ S im p Γ (A). N otic e that sinc e V no long er

is (nec essarily ) simple, w e do not k now that this map is topolog ic ally su rjec tiv .

N ow , Bs (Γ ) is c ommu tativ e, so the k-su b -alg eb ra Cs (Γ ) ⊂ Bs (Γ ) g enerated b y the

elements {xi ,q }i= 1 ,..,d ; p ,q = 1 ,..,n is c ommu tativ e. W e hav e a morphism,

p





Is (Γ ) : A → Cs (Γ ) ⊗k Mn (k) = Mn (Cs (Γ )).



M oreov er, these Cs (Γ ) defi ne a presheaf, C(Γ ), on the J ac ob son topolog y of

S im p Γ (A). T he rank n free Cs (Γ )-modu les w ith the A-ac tions g iv en b y Is (Γ ), g lu e

tog ether to form a loc ally free C(Γ )-M odu le E(Γ ) on S im p Γ (A), and the morphisms

Is (Γ ) indu c e a morphism of alg eb ras,



I(Γ ) : A → E nd C(Γ) (E(Γ )).

GEOMETRY OF TIME-SPACES. 25



As for every V ∈ S im p Γ (A), E nd A (V ) = k, th e c om m u ta tor of A in H A (V )c o m ⊗k

E nd k (V ) is H A (V )c o m . T h e m orp h ism ,



ζ(V ) : H A (V )c o m → HH 0 (A, H A (V )c o m ⊗k E nd k (V ))



is th erefore a n isom orp h ism , a n d w e m a y a ssu m e th a t th e c orresp on d in g m orp h ism ,



ζ : C(Γ ) → HH 0 (A, E nd C(Γ) (E(Γ )))



is a n isom orp h ism of sh ea ves. F or a ll V ∈ D(s) ⊂ S im p Γ (A) th ere is a n a tu ra l

p rojec tion ,



κ := κ(Γ ) : Cs (Γ ) ⊗k Mn (k) → H A(n) (V )c o m ⊗k E nd k (V ) Mn (H A(n) (V )c o m ),



w h ich , c om p osed w ith Is (Γ ) is th e n a tu ra l h om om orp h ism ,



A − → H A(n) (V )c o m ⊗k E nd k (V ).



κ d efi n es a set th eoretic a l m a p ,



t : S im p Γ (A) − → S p e c(C(Γ )),



a n d a n a tu ra l su rjec tive h om om orp h ism ,

ˆ

C(Γ )t(V ) → H A(n) (V )c o m .



C a teg oric a l p rop erties im p lies, a s u su a l, th a t th ere is a n oth er n a tu ra l m orp h ism ,

ˆ

ι : H A(n) (V ) → C(Γ )t(V ) ,



w h ich c om p osed w ith th e form er is th e ob viou s su rjec tion , a n d su ch th a t th e in d u c ed

c om p osition ,

ˆ

A − → H A(n) (V )c o m ⊗k E nd k (V ) → C(Γ )t(V ) ⊗k E nd k (V ),



is I(Γ ) form a liz ed a t t(V ). F rom th is, a n d from th e d efi n ition of C(Γ ), it follow s

th a t ι is su rjec tive, su ch th a t for every V ∈ S im p Γ (A) th ere is a n isom orp h ism

H A(n) (V )c o m ˆ

C(Γ )t(V ) . F or V ∈ S im p Γ (A) th ere is a lso a n a tu ra l c om m u ta tive

d ia g ra m ,



Z A(n) / C(Γ )





 

A(n) / E nd C(Γ) (E(Γ ))





 

H A(n) (V ) ⊗k E nd k (V ) ˆ

/ C(Γ ) ⊂t(V ) (n) ⊗k E nd k (V )



F orm a lly a t a p oin t V ∈ S im p Γ (A), w e h a ve th erefore p roved th a t th e loc a l, c om m u -

ta tive stru c tu re of Sim p Γ (A) (a s A or A(n)-m od u le), a n d th e c orresp on d in g loc a l

stru c tu re of S p e c(C(Γ )) a t V , c oin c id e. W e h a ve a c tu a lly p roved th e follow in g ,

26 OLAV ARNFINN LAUDAL



Theorem 2.15. The topological space SimpΓ (A), w ith the J acob son topology , to-

gether w ith the sheaf of com m u tativ e k-algeb ras C(Γ ) d efi n es a schem e stru ctu re on

SimpΓ (A), an d a m orphism ,

Spe c(C(Γ )) → SimpΓ (A).

M oreov er, Spe c(C(Γ )) con tain s an open su b schem e, ´ tale ov er Simpn (A), an d there

e

is a m orphism ,

π(Γ ) : SimpΓ (A) → Spe c(Z A(n)),

ex ten d in g the n atu ral m orphism ,

π0 : Simpn (A) → Spe c(Z A(n)).



P roof. A s in T h e o re m (2 .1 3 ) w e p ro v e th a t if v = t(V ), w ith V ∈ Simpn (A) ⊆

SimpΓ (A), th e n th e re e x ists a n o p e n su b sch e m e o f Spe c(C(Γ )) c o n ta in in g o n ly

sim p le m o d u le s o f d im e n sio n n. If v is in d e c o m p o sa b le s w ith E nd A (V ) = k w e

m a y lo o k a t th e h o m o m o rp h ism o f C(Γ )-m o d u le s,

E nd A (C(Γ )) ⊗ E nd k (V ) −→ E nd A (V ) = k.

C le a rly th e re is a n o p e n n e ig h b o rh o o d o f v in Spe c(C(Γ )) c o n ta in in g o n ly in d e c o m -

p o sa b le s o f d im e n sio n n.



T h e se m o rp h ism s π(Γ ) a re o u r c a n d id a te s fo r th e p o ssib ly d iff e re n t c o m p le -

tio n s o f Simpn (A). N o tic e th a t fo r W ∈ Spe c(C(n)) − U (n), th e fo rm a l m o d -

u li H A (W ) is n o t a lw a y s p ro -re p re se n tin g . If W is se m i-sim p le , b u t n o t sim p le

th e n E nd A (W ) = k. T h e c o rre sp o n d in g m o d u la r su b stra tu m w ill, lo c a lly , c o r-

re sp o n d to th e se m i-sim p le d e fo rm a tio n s o f W , th u s to a c lo se d su b sch e m e o f

Spe c(C(n)) − Un ⊂ Spe c(C(n)). T h is fo llo w s fro m th e fa c t th a t th e su b stra -

tu m o f m o d u la r d e fo rm a tio n s o f a n y se m isim p le (b u t n o t sim p le ) m o d u le V w ill

h a v e a ta n g e n t sp a c e e q u a l to th e in v a ria n t sp a c e o f th e a c tio n o f th e E nd k (V )

o n E xt1 (V, V ), w h ich m u st b e th e su m o f th e ta n g e n t sp a c e s o f th e d e fo rm a tio n

A

sp a c e s o f th e sim p le c o m p o n e n ts o f V .

Spe c(C(n)) is, in a se n se , a c o m p a c tifi c a tio n o f Un . It is, h o w e v e r n o t th e c o rre c t

com pletion o f U (n). In fa c t, th e p o in ts o f Spe c(C(n)) − U (n) m a y c o rre sp o n d to

se m i-sim p le m o d u le s, w h ich d o n o t d e fo rm in to sim p le n -d im e n sio n a l m o d u le s. W e

sh a ll in §4 re tu rn to th e stu d y o f th e (n o tio n o f) c o m p le tio n , in c o n n e c tio n w ith th e

p ro c e ss o f d ecay a n d creation o f p a rtic le s. D e c a y o c c u r, a t in fi n ity in Simpn (A),

se e th e In tro d u c tio n .

H ilbert schem es an d S trin gs. A b o v e w e h a v e stu d ie d m o d u li sp a c e s o f re p re -

se n ta tio n s o f fi n ite ly g e n e ra te d k-a lg e b ra s. W e m ig h t a s w e ll h a v e stu d ie d th e

H ilb e rt fu n c to r, HA , o f th e a lg e b ra A, o r th e m o d u li sp a c e M(A; R), o f m o r-

p h ism s, κ : A → R, fo r fi x e d a lg e b ra s, A a n d R. T h e d iff e re n c e is th a t w h e re a s

fo r fi n ite n, th e se t Simpn (A) h a s a n ic e , fi n ite d im e n sio n a l sch e m e stru c tu re ,

th is is, in g e n e ra l, n o lo n g e r tru e fo r th e se ts, HA o r M(A; R). If, h o w e v e r, R

n

is A rtin ia n o f le n g th n, th e n th e c o rre sp o n d in g H ilb e rt sch e m e , HA , d o e s e x ist

a n d h a s a n ic e stru c tu re , b o th a s c o m m u ta tiv e a n d a s n o n -c o m m u ta tiv e sch e m e .

In p a rtic u la r, th e to y m o d e l o f re la tiv ity th e o ry , re fe rre d to in th e in tro d u c tio n ,

is m o d e le d o n M(k[x1 , x2 , x3 ], k 2 ), i.e . o n th e se t o f su rje c tiv e h o m o m o rp h ism s

k[x1 , x2 , x3 ] → R = k 2 . W e sh a ll n o w e n rich th e se stru c tu re s so m e w h a t b y in tro -

d u c in g th e n o tio n o f a strin g.

GEOMETRY OF TIME-SPACES. 27



Definition 2.16. A general string, a g-string, is an algebra R to geth er w ith a p air

o f P h -p o ints, i.e. a p air o f h o m o m o rp h ism s i : P h R → k(pi ), co rresp o nd ing to tw o

p o ints k(pi ) ∈ S im p1 (R) eac h o u tfi tted w ith a tangent ξi .

W e m ig h t h a v e c o n sid e re d a n y tw o p o in ts k(pi ) ∈ S im pn (P h R), b u t sin c e th e

m a in p ro p e rtie s o f th e g -strin g s w ill b e e q u a lly w e ll u n d e rsto o d re stric tin g to th e

c a se n = 1 , w e sh a ll p o stp o n e th is g e n e ra liz a tio n .

F o r a n y g -strin g , c o n sid e r th e no n-co m m u tativ e tangent sp ace o f th e th e p a ir o f

p o in ts,

T (R, p1 , p2 ) := E xt1 h R (p1 , p2 ).

P

W e sh a ll c a ll it th e sp ace o f tensio ns, b e tw e e n th e tw o p o in ts o f th e strin g .

C o n sid e r th e sp a c e S tr ingg (A) o f g − s tr ings in A, i.e . th e sp a c e o f iso m o rp h ism

c la sse s o f a lg e b ra h o m o m o rp h ism s κ : A → R w h e re R is a g -strin g , a n d w h e re

iso m o rp h ism s sh o u ld c o rre sp o n d to iso m o rp h ism s o f th e g -strin g , th u s c o n se rv in g

th e tw o P h R-p o in ts.

A n y g -strin g in A, κ : A → R, in d u c e s a u n iq u e c o m m u ta tiv e d ia g ra m o f a lg e b ra s,

κ /R

A

i iR

 P hκ 

P hA / P h R.

T h e v o n N e u m a n n c o n d itio n im p o se d o n a strin g κ, is n o w th e fo llo w in g ,

i ◦ P h κ ◦ d = κ∗ ξi =: ξ i = 0 , i = 1 ∨ i = 2,

w h ich , if xj , j = 1 , .., n a n d σl , l = 1 , .., p a re p a ra m e te rs o f A re sp e c tiv e ly R, is

e q u iv a le n t to th e c o n d itio n ,

∂ xj

(pi ) = 0 , j = 1 , ..., n, l = 1 , .., p, i = 1 ∨ 2.

∂σ l

N o tic e a lso th a t, sin c e a n y d e riv a tio n ξ ∈ D e r k (A, R) h a s a n a tu ra l liftin g to a

d e riv a tio n ξ ∈ D e r k (P h A, P h R) d e fi n e d b y sim p ly p u ttin g ξ(a) = d(ξ(a)), w e

fi n d , u sin g th e g e n e ra l m a ch in e ry o f d e fo rm a tio n s o f d ia g ra m s, se e [L a 0 ], th a t a n y

fa m ily o f m o rp h ism s κ in d u c e s a fa m ily o f th e a b o v e d ia g ra m . If τk , k = 1 , ..., d

a re p a ra m e te rs o f su ch a fa m ily , M = S pe c(M ), th e n dτi ∈ P h M c o rre sp o n d s to a

d e riv a tio n , τi ∈ D e r k (A, R), a n d th e re fo re to ta n g e n ts ξ i , i = 1 , 2, o f S im p1 (A) a t

th e tw o p o in ts k(pi ). T h e D irich le t c o n d itio n o n th e strin g is n o w ,

ξ i = 0 , i = 1 ∨ 2,

w h ich is e q u iv a le n t to th e c o n d itio n ,

∂ xj

(pi ) = 0 , j = 1 , ..., n, l = 1 , .., p, i = 1 ∨ 2.

∂ τl

v

T h e se c o n d itio n s w ill d e fi n e n e w m o d u li sp a c e s w h ich w e sh a ll c a ll S tr ingRN (A)

D

a n d S tr ingR (A), re sp e c tiv e ly . In th e a ffi n e c a se th e stru c tu re o f th e se sp a c e s is a

p ro b le m , h o w e v e r w e m a y o f c o u rse d o e v e ry th in g a b o v e fo r A a n d R re p la c e d b y

p ro je c tiv e sch e m e s, a n d th e n a ll th e m o d u li sp a c e s e x ist a s c la ssic a l sch e m e s. W e

sh a ll n o t v e n tu re in to th is ” strin g -th e o ry ” , b u t se e th e n e x t §, w h e re w e w o rk o u t

a g e n e ra l m e th o d fo r in tro d u c in g d y nam ic s o n m o d u li sp a c e s o f re p re se n ta tio n s o f

a sso c ia tiv e k-a lg e b ra s. T h e sa m e m e th o d m ig h t b e a p p lic a b le to th e m o d u li sp a c e

o f strings in th e a b o v e se n se .

28 OLAV ARNFINN LAUDAL



Example 2.17. (i) Let us go back to Example (1.1)(ii). It follows that the string

of d imension 0 , R = k 2 , P h (R) = k /((x2 − r2 ), (xd x + d xx)), has uniq ue

points, k(±r). T he space of tensions is of d imension 1, the v on N eumann cond ition

is automatically satisfi ed , and the mod uli space of k 2 -strings in A = k[x1 , x2 , x3 ]

is nothing but H := S pe ck[t1 , ..., t6 ] − ∆. If we consid er the string with R =

k[x]/(x2 ), P h R = k[x, d x]/(x2 , (xd x+d xx), then we see that there is just one point

of R, but a line of point for P h R, all correspond ing to x = 0 in R. T herefore there is

a 2-d imensional space of strings with the same R. C ompare this with the blow-up H, ˜

see [La 6 ]. (ii) In d imension 1 the simplest closed string is giv en by , R = k[x, y]/(f ),

with f = x2 +y 2 −r2 , such that P h R = k /(f, [x, y], d [x, y], df ), and

with the two points, i : P h R → k(pi ), d efi ned by the actions on k(pi ) := k, giv en

by , xi , yi , (d x)i , (d y)i , i = 1, 2. It is easy to see that the v ectors, ξi := ((d x)i , (d y)i )

are tangent v ectors to the circle at the points pi , and if p1 = p2 we fi nd that

E xt1 h R (k(p1 ), k(p2 ) = k. T he v on N eumann cond ition is, ξi = 0 , i = 1 ∨ i = 2,

P

∂x ∂y

and this clearly means that = = 0 at one of the points pi . T he 1-d imensional

∂σ ∂σ

op en string is now left as an exercise.

§3 . G eo metry o f time-spac es an d th e g en eral d y n amic al law . G iv en a

fi nitely generated k-algebra, and a natural number n, we hav e in §2 constructed a

scheme S im pn (A), and and a v ersal family ,



˜ ˜

ρ : A(σ) → E nd U (n) (V )



e

d efi ned on an ´ tale cov ering U (n) of S im pn (A).

U (n) is an open subscheme of an affi ne scheme S pe c(C(n)), and the v ersal family

is, in fact, d efi ned on C(n).

W e would now like to use this theory for the k-algebra P h ∞ (A) of §1. H owev er,

P h ∞ (A) is rarely of fi nite ty pe. W e shall therefore intod uce the notion of dy na m ica l

stru ctu re, and the order of a d y namical structure, to red uce the problem to a

situation we can hand le. T his is also what phy sicists d o, they inv oke a parsimony

principle, originally proposed by F ermat, and later by M aupertuis, with exactely

this purpose, red ucing the preparation need ed to be able to see ahead , see (1.2).

D efi n itio n 3 .1. A dy na m ica l stru ctu re, or sy stem , σ, is a tw o-sided δ-idea l (σ) ⊂

P h ∞ (A), su ch th a t

A(σ) := P h ∞ (A)/(σ)

is of fi nite ty p e. A dy na m ica l stru ctu re is of order n if th e ca nonica l m orp h ism ,



σ : P h (n−1) → A(σ)



is su rjectiv e. If A is genera ted b y th e coordina te fu nctions, {ti }i= 1,2,...,d a ny dy -

na m ica l sy stem of order n is defi ned b y a F orce L a w , i.e. a sy stem of eq u a tions,



δ n tp = Γp (ti , d tj , d 2 tk , .., d n−1 tl ), p = 1, 2, ..., d .



P u t,

A(σ) := P h ∞ (A)/(δ n tp − Γp )

w h ere σ := (δ n tp − Γp ) is th e tw o-sided δ-idea l genera ted b y th e defi ning eq u a tions

of σ. O b v iou sly δ indu ces a deriv a tion δσ ∈ D e rk (A(σ)), a lso ca lled th e D ira c

deriv a tion, a nd u su a lly ju st denoted δ.

GEOMETRY OF TIME-SPACES. 29



Notice that if σi , i = 1 , 2 , are tw o ord er n d y n am ical sy stem s, then w e m ay w ell

hav e,

A(σ1 ) A(σ2 ) P h (n−1) (A/σ∗ ),

as k-alg eb ras, see the In trod u ction .

The general dynamical law and the evolution operator.

F or an y in teg er n ≥ 1 con sid er the schem es S im p n (A(σ)) an d S p e c(C(n)), an d

the corresp on d in g v ersal fam ily ,



˜ ˜

ρ : A(σ)) → E nd S p e c (C(n)) (V ) ⊆ Mn (C(n)).



T he D irac d eriv ation δ ∈ D e r k (A(σ), A(σ)) d efi n es, as ex p lain ed ab ov e, a d istru b u -

tion on S im p n (A(σ)). T he reason w hy the D irac d eriv ation δ, d oes n ot n ecessarily

d efi n e a u n iq u e v ector-fi eld is, of cou rse, that the stru ctu re m ap s of the sim p le m od -

u les m ig ht b e scaled b y a n on -z ero elem en t of the fi eld k. H ow ev er, on ce w e hav e

chosen a v ersal fam ily for the m od u li sp ace S im p n (A(σ)), d efi n ed on S p e c(C(n)),

the D irac d eriv ation δ in d u ces, b y com p osition , an elem en t,

˜ ˜

δ ∈ D e r k (A(σ), E nd C(n) (V )).



w hich ob v iou sly in d u ces a w ell d efi n ed v ector fi eld ξ ∈ ΘU (n) , in the d istrib u tion

d efi n ed b y δ. Now , to an y v ectorfi eld ξ of S p e c(C(n)), i.e. for an y d eriv ation

ξ ∈ D e r k (C(n)), there is a u n iq u e elem en t,

˜

ξ ∈ D e r k (A(σ), E nd C(n) (V )),



d efi n ed b y ,

ρ

ξ (a) = ξ(˜(a)),

˜

w here w e hav e id en tifi ed ρ(a) w ith an elem en t of Mn (C(n)). M oreov er, this d eriv a-

tion is a liftin g of ξ, i.e. it in d u ces the sam e v ector fi eld on S p e c(C(n)) as ξ.T o see

this, con sid er the classical isom orp hism ,



E x t1 ˜ ˜ ˜

C(n)⊗A(σ) (V , V ) D e r k (C(n) ⊗ A(σ), E nd k (V ))/ ∼



w here the eq u iv alen ce relation is g iv en in term s of triv ial d eriv ation s. R ecall that

˜ C(n) ⊗k V , as C(n)-m od u le. T herefore

V

˜ ˜ ˜

E x t1 (V , V ) = D e r k (C(n), E nd k (V ))/ ∼= 0 ,

C(n)



˜

i.e. ev ery elem en t of D e r k (C(n), E nd k (V )) is an in n er d eriv ation . Notice also that

the k ern el of the n atu ral su rjectiv e m ap ,

˜ ˜

D e r k (C(n) ⊗ A(σ), E nd k (V )) → D e r k (C(n), E nd k (V ))



is,

˜

D e r k (A(σ), E nd C(n) (V )).

˜ ˜ ˜

In fact, if δ ∈ D e r k (C(n) ⊗ A(σ), E nd k (V )) m ap s to z ero in D e r k (C(n), E nd k (V )),

˜ ˜ ˜

then cδ(a) = δ(c)a + cδ(a) = δ(c ˜ ˜ ˜

˜ ⊗ a) = δ(a)c + aδ(c) = δ(a)c, sin ce c ⊗ a =

(c ⊗ 1 )(1 ⊗ a) = (1 ⊗ a)(c ⊗ 1 ). T herefore,



E x t1 ˜ ˜ ˜

C(n)⊗A(σ) (V , V ) D e r k (A(σ), E nd C(n) (V ))/ ∼ .

30 OLAV ARNFINN LAUDAL



˜

By construction of C(n), a nd of th e v e rsa l fa m ily, ρ in g e ne ra l, w e h a v e for a ny a ∈

A(σ), a nd a ny p oint V ∈ U (n), corre sp ond ing to a m a x im a l id e a l (ti ), i = 1 , 2, ..., r

of C(n),

r

˜

ρ(a)(v) = ρV (a)(v) + ti ψi (a, v) + h ig h e r ord e r te rm s in ti ,

i=1



w h e re ψi ∈ D er k (A(σ), E nd k (V )), i = 1 , 2, ..., r a re re p re se nta tiv e s of a b a sis for,



E x t1 (V, V )

A(σ) D er k (A(σ), E nd k (V ))/ ∼ .



S up p ose δi ∈ D er k (C(n), C(n)) is a d e riv a tion th a t, m od ulo th e m a x im a l id e a l (ti )

se nd s ti to 1 , a nd a ll oth e r tj to 0 , th e n th e d e riv a tion



˜

δi ∈ D er k (A(σ), E nd C(n) (V )),



d e fi ne d a b ov e , com p ose d w ith th e h om om orp h ism C(n) → C(n)/(ti ) = k(V ), is

just, ψi . T h is , tog e th e r w ith th e na tura l isom orp h ism p rov e d a b ov e ,



θU (n) E x t1 ˜ ˜

C(n)⊗A(σ) (V , V )|U (n),



˜

p rov e s th a t δ −ξ ∈ D er k (A(σ), E nd C(n) (V )) is a a n inne r d e riv a tion g iv e n b y som e

e le m e nt Q ∈ Mn (C(n)). P ut, [δ] := ξ . W e h a v e p rov e d th e fund a m e nta l re sult:

˜

T h e o re m 3 .2 . As operators on V , w e m u st h av e,



δ = [δ] + [Q, −].



˜

T h is m eans th at for ev ery a ∈ A(σ), consid ered as an elem ent ρ(a) ∈ Mn (C(n)),

˜

δ(a) ac ts on V as

˜ ρ ˜

ρ(δ(a)) = ξ(˜(a)) + [Q, ρ(a)].



T h is Q, th e H am iltonian of th e syste m , is, in th e sing ula r ca se , w h e n [δ] = 0 , a lso

¨

ca lle d th e D ira c op e ra tor, a nd som e tim e s d e note d d elta slash ed , se e e .g . [S ch uck e r],

or oth e r te x ts on C onne s’ sp e ctra l trip p le s. In fa ct, a spec tral tripple is com p ose d of

˜

a v e ctor sp a ce lik e V , tog e th e r w ith a D ira c op e ra tor, lik e Q, a nd a com p le x ifi ca tion

e tc.

If [δ] = 0 , it is a lso e a sy to se e th a t w h a t w e h a v e ob se rv e d im p lie s th a t H e ise n-

o

b e rg ’s a nd S ch r¨ d ing e r’s w a y of d oing q ua ntum m e ch a nics a re stricte ly e q uiv a le nt.

In line w ith our g e ne ra l p h ilosop h y, w e sh a ll consid e r ξ, or δ a s m e a suring tim e

on S p ec(C(n)) or S im p n (A(σ)).

A ssum e from now on th a t k = R, th e re a l num b e rs, a nd th a t our constructions g o

th roug h , a s if k w e re a lg e b ra ica lly close d . L e t v(τ 0 ) ∈ S im p n (A(σ)) b e a n e le m e nt,

a n ev ent. S up p ose th e re e x ist a n inte g ra l curv e γ of ξ th roug h v(τ 0 ) ∈ S im p 1 (C(n)),

e nd ing a t v(τ 1 ) ∈ S im p 1 (C(n)), g iv e n b y th e a utom orp h ism s e(τ ) := ex p (τ ξ), for

τ ∈ [τ0 , τ1 ] ⊂ R. T h e m a x im um τ for w h ich th e e nd p oint, t, of γ is in S im p n (A(σ))

sh ould b e ca lle d th e lifetim e of th e p a rticle . W e sh a ll se e th a t it is re la tiv e ly e a sy to

com p ute th e se life tim e s, w h e n th e fund a m e nta l v e ctor fi e ld ξ h a s b e e n com p ute d .

T h is, h ow e v e r, is ce rta inly not so e a sy, se e th e e x a m p le s (3 .4 )-(3 ,8 ).

GEOMETRY OF TIME-SPACES. 31



˜

Let now ψ(τ0 ) ∈ V (v0 ) V b e a (c la ssic a lly c onsid ered ) sta te of ou r quantum

sy ste m, a t th e tim e τ0 , a nd c onsid er th e (u ni)v ersa l fa m ily ,



ρ : A(σ) −→

˜ ˜

E nd C(n) (V )

wh ere S im p n (A(σ)) ⊆ S im p 1 (C(n)). W e sh a ll c onsid er A(σ) a s ou r ring of o b -

se rv ab le s.

W h a t h a p p ens to ψ(τ0 ) ∈ V (0 ) wh en time p a sses from τ0 to τ , a long γ? T h is

is ob v iou sly a q u estion th a t h a s to d o with wh eth er we ch oose to c onsid er th e

o

th e H eisenb erg or th e S ch r¨ d ing er p ic tu re. In fa c t, if we c onsid er th e form a l fl ow

e x p (tδ) d efi ned on th e ring of ob serv a b les, th en p u tting ,

u(τ ) := e x p (τ ξ ),



wh ere,

ξ

˜

:= ξ + Q ∈ E nd k (V ),

˜

we ob ta in for ev ery ψ ∈ V , a nd ev ery a ∈ A(σ), th a t th e eq u a tion,

˜

u(τ )(˜(e x p (−τ δ)(a))(ψ)) = ρ(a)(u(τ )(ψ))

ρ

h old s form a lly , a t lea st u p to fi rst ord er. In fa c t, u p to ord er one, in τ , th e left

h a nd sid e is eq u a l to

˜ ˜

ρ(a)(ψ) − τ ρ(δ(a)(ψ) + τ ξ(˜(a)(ψ)) + τ Q˜(a)(ψ),

ρ ρ

a nd th e rig h t h a nd sid e is,

˜ ˜ ˜

ρ(a)(ψ) + τ ρ(a)(ξ(ψ)) + τ ρ(a)(Q(ψ)).

˜

N otic ing th a t ξ(˜(a)(ψ)) = ξ(˜(a))(ψ) + ρ(a)(ξ(ψ)), a nd u sing (3 .2 ) we fi nd th a t

ρ ρ

th e two sid es a re eq u a l.

˜

T h is m ea ns th a t e x p (τ δ) k eep s V fi x ed with in its c onju g a te c la ss, u p to fi rst

ord er in τ . T h u s, a n elem ent ψ ∈ V ˜ wh ich is fl at with resp ec t to th e c onnec tion

ξ , a b ov e γ, h a s th e p rop erty th a t,



˜

ρ(δ(a))ψ = ξ (˜(a)(ψ)),

ρ

for a ll a ∈ A(σ).

˜

It is th erefore rea sona b le to c onsid er a ny fl a t sta te, ψ(t) ∈ V , a s th e tim e d e-

v elop m ent of ψ(0 ) ∈ V (0 ). C lea rly , th e fl a t sta tes ψ ∈ V ˜ , a re solu tions of th e

d iff erentia l eq u a tion,

∂ψ

ξ(ψ) = −Q(ψ), i.e . = −Q(ψ).

∂τ

wh ich , if we a c c ep t th a t tim e is th e p a ra m eter τ of th e integ ra l c u rv e γ, is th e

o

S ch r¨ d ing er eq u a tion.

N otic e th a t, in th e c la ssic a l q u a ntu m -th eoretic a l c a se, one work s with one fi x ed

rep resenta tion, c orresp ond ing to wh a t we h a v e c a lled a sing u la r p oint of ξ. T h is

im p lies th a t we a re look ing a t a rep resenta tion V with ξ(v) = 0 , a nd so we h a v e no

time . W h a t we c a ll tim e is th en th e p a ra m eter of th e one-p a ra m eter a u tom orp h ism

o

g rou p u(τ ) := e x p (τ Q) a c ting on V . T h is a lso lea d s to a S ch r¨ d ing er eq u a tion,

a nd to th e nex t resu lt, p rov ing th a t ψ is c om p letely d eterm ined , a long a ny integ ra l

c u rv e γ b y th e v a lu e of ψ(τ0 ), for a ny τ0 ∈ γ.

32 OLAV ARNFINN LAUDAL



Theorem 3.3. The evolution operator U (τ0 , τ1 ) that c hang es the state ψ(τ0 ) ∈

˜ ˜

V (v0 ) into the state ψ(τ1 ) ∈ V (v1 ), w here τ1 − τ0 is the leng th of the integ ral c urve

γ connec ting the tw o points v0 and v1 , i.e. the tim e passed , is g iven b y ,



ψ(τ1 ) = U (τ0 , τ1 )(ψ(τ0 )) = e xp [ Q(τ )d τ ] (ψ(τ0 )),

γ





w here e xp γ is the non-com m utative version of the ord inary ac tion integ ral, essen-

tially d efi ned b y the eq uation,



e xp [ Q(τ )d t] = e xp [ Q(τ )d τ ] ◦ e xp [ Q(τ )d τ ]

γ γ2 γ1





w here γ is γ1 follow ed b y γ2 .

o

P roof. T h is is a w e ll k n o w n c o n se q u e n c e o f th e S ch r¨ d in g e r e q u a tio n a b o v e . In

c la ssic a l q u a n tu m th e o ry o n e u se s a c hronolog ical operator τ , to k e e p tra ck o f th e

interm ed iate tim e -ste p s th a t, in o u r c a se , a re w e ll d e fi n e d b y th e in te g ra l c u rv e

γ, th e e x iste n c e o f w h ich w e a ssu m e . T h e fo rm u la a b o v e is re la te d to w h a t th e

p h y sic ists c a ll th e D y so n se rie s, se e [W e in b e rg ], V o l I, C h a p . 9 , o r [E lb a z ], C h a p itre

6 . S in c e w e h a v e g iv e n th e re a l c u rv e γ p a ra m e triz e d b y τ w e m a y lo o k a t γ a s a

c lo se d in te rv a l o f R, I := [0 , τ ]. S u b d iv id e I in to m e q u a l in te rv a ls, [i∆τ, (i+ 1 )∆τ ],

o

a n d se e th a t th e S ch r¨ d in g e r e q u a tio n g iv e s, fo rm a lly ,



ψ((i + 1 )∆τ ) = e xp (∆τ Q)(ψ(i∆τ )).



W ritin g o u t th e p o w e r se rie s in ∆τ , a n d su m m in g u p w e fi n d th e fo rm u la a b o v e .





P ath integ rals and Q uantum theory .. N o tic e th a t w e fi n d th e sa m e fo rm u la s if

˜ ˜

w e e x te n d th e a c tio n o f A(σ) to VC := V ⊗R C. T h is is w h a t tu rn s o u t to b e th e

in te re stin g c a se in q u a n tu m p h y sic s. It is e a sy to se e th a t if A = k[x1 , ..., xd ] ⊂ A(σ)

is a p o ly n o m ia l a lg e b ra , a n d σ is a se c o n d o rd e r fo rc e -la w , th e n if w e h a v e ch o se n

a v e rsa l fa m ily ,

ρ : A(σ) → E nd C(n) (V )

˜ ˜



fo r th e sim p le n-d im e n sio n a l re p re se n ta tio n s, w e o b ta in a n o th e r, c o m p le x ifi e d , v e r-

sa l fa m ily ,

ρC : A(σ) → E nd C(n) (VC )

˜ ˜



˜ ˜ ˜

w ith e x a c tly th e sa m e fo rm a l p ro p e rtie s b y d e fi n in g , ρC (xi ) = ρ(xi ), ρC (d xi ) =

ı˜(d xi ), a n d p u ttin g ξC = ıξ, QC = ıQ.

ρ

N o w , sin c e th e confi g uration-space c o o rd in a te s xi c o m m u te , w e m a y fi n d ra tio n a l

se c tio n s

˜

|xν (t) >∈ VC , ν = 1 , ..., n,

th a t a re e ig e n v e c to rs fo r a ll xi , su ch th a t a p o in t in confi g uration space is g iv e n b y

th e n p o ssib ilitie s, (x1,ν (t), ..., xd,ν (t)), w h e re ,



ρ(xi )(|xν (t) >) = xi,ν (t)|xν (t) > .

˜

GEOMETRY OF TIME-SPACES. 33



Now, pick a point t0 ∈ U (n), and assu m e we h av e com pu te d th e inte g ral cu rv e γ

th rou g h t0 , e nd ing at t1 , param e triz e d b y τ . T h e e v olu tion ope rator U (τ0 , τ1 ) acts

u pon e ach |xν (t0 ) >, ν = 1 , ..., n. T h e re su lt will h av e th e form ,



U (τ0 , τ1 )(|xνi (t0 ) >) = γνi ,νj (τ )|xνj (t1 ) >,

j= 1,..,n



wh e re e ach γνi ,νj (τ ) is a kind of action inte g ral re late d to th e classical L ag rang ian.

H owe v e r, we h av e to b e care fu l not to d raw any ph y sical conclu sion from th is. In

fact, we h av e pre pare d th e object V (t0 ) too we ll, h av ing assu m e d th at we know,

tog e th e r with th e action of th e confi g u ration space coord inate s, xi also th e action

of th e ir m om e nta d xi . S o to ob tain p roba bilities, for fi nd ing |xν1 (t1 ) > at tim e τ

corre spond ing to th e point t1 , h av ing pre pare d th e ob je ct as |xν0 (t0 ) > at tim e 0 ,

corre spond ing to th e point t0 , we h av e , not only to introd u ce H e rm itian norm s on

th e v e ctorspace VC b u t also to inte g rate on all possib le m om e nta, i.e . on all cu rv e s

γ, starting at th e point in confi g u ration space corre spond ing to |xνi (t0 ) > le ad ing

to th e sam e e nd re su lt |xνj (t1 ) >. T h is se e m s to le ad to a kind of g e ne raliz e d

F e y nm an’s inte g ral, wh ich I sh all h ope fu lly re tu rn to in a late r pape r. F or a g ood

e x position, for m ath e m aticians, of path inte g rals, se e [F ad d e e v ].

P la n ck ’s con sta n t(s), E h ren fest’s th eorem , a n d th e F ock rep resen ta tion . In [L a

5 ] we tre ate d th e case of a conse rv ativ e sy ste m , i.e . wh e re th e v e ctor fi e ld ξ in

S im p n (A(σ) is sing u lar, i.e . v anish e s, at th e point v ∈ S im p n (A(σ) corre spond ing

to th e re pre se ntation V , and wh e re th e re fore th e H am iltonian Q is b oth th e tim e

and e ne rg y ope rator, at th e sam e ” tim e ” . S e e e x am ple s (3 .6 ) and (3 .7 ) wh e re we

sh ow h ow to com pu te th e se sing u laritie s in som e classical case s.

W e fou nd , in th is situ ation, se e [L a 6 ], or §1 , th at th e re is a notion of P la n ck ’s

con sta n t , with th e ord inary prope rtie s.

T h is is also tru e in g e ne ral. In fact, since



[Q, ρ(−)] = ρδ − [δ]˜ : A(σ) −→

˜ ˜ ρ ˜

E nd C(n) (V )



is a d e riv ation, we sh ow th at th e se t,



˜

Λ (σ) := {λ ∈ C(n)|∃ fλ ∈ A(σ), fλ = 0 , [Q, ρ(δ(fλ ))] = λ˜(fλ )},

ρ



is a g e ne raliz e d ad d itiv e m onoid , i.e . if for λ, λ ∈ Λ (σ) th e prod u ct fλ fλ is non-

triv ial, th e n λ + λ ∈ Λ (σ).

L e t l ∈ k b e ” g e ne rators” of Λ (δ). T h e se are ou r P lanck’s constants, se e

˜

e x am ple s (3 .7 ) and (3 .8 ). Now, assu m e th e re e x ists a C(n)-m od u le b asis {ψi }i∈I of

se ctions of V ˜ = C(n) ⊗ V , form e d b y e ig e nfu nctions for th e H am iltonian, i.e . su ch

th at

˜ ˜

Q(ψi ) = κi ψi , i ∈ I ,

˜ ˜

wh e re κi ∈ C(n). A n e le m e nt su ch as ψi ∈ V is u su ally consid e re d as a p u re sta te,

with en erg y κi ∈ C(n), d e pe nd ing on tim e , i.e . d e pe nd ing on τ , th e le ng th along

˜

th e inte g ral cu rv e γ. It is also consid e re d as as an elem en ta ry p a rticle (since V is,

b y assu m ption, sim ple ). A s in §1 we fi nd ,

˜ ˜ ˜ ˜

λ˜(fλ )(ψi ) = Q(˜(fλ )(ψi )) − ρ(fλ )(Q(ψi ))

ρ ρ

˜ ˜ ˜

= Q(˜(fλ )(ψi )) − κi ρ(fλ )(ψi )

ρ

34 OLAV ARNFINN LAUDAL



implying,

ρ ˜ ρ ˜

Q(˜(fλ )(ψi )) = (κi + λ)˜(fλ )(ψi ).

˜ ˜

B y a ssu mtio n, if ρ(fλ )(ψi ) = 0 it mu st b e a n e ige nv e c to r o f Q, w ith e ige nv a lu e , sa y

κj = κi + λ. It fo llo w s th a t w e h a v e ,



Λ (σ) ⊂ {κj − κi | i, j ∈ I}.



T o pro v e th a t th e tw o se ts a re e q u a l w e ne e d so me e x tra c o nd itio ns o n th e na tu re

˜ ρ ˜

o f A(σ) a nd ρ. If {˜(fλ }λ ge ne ra te E nd C(n) (V ), th e n th e e q u a lity mu st h o ld ,

sinc e th e n {˜(fλ )(ψ(0 )}λ mu st ge ne ra te V

ρ ˜ a s C(n)-mo d u le , a nd th e re fo re c o nta in

mu ltiple s o f a ll ψj , so th a t a ny κl mu st b e e q u a l to κ0 + λ fo r so me λ.

˜

N o tic e th a t if goes to 0, me a ning th a t [Q, ρ(a)] = 0 , fo r a ll a ∈ A(σ), th e n a ll

a ∈ A(σ) mu st c o mmu te w ith Q, a nd so a c t d ia go na lly o n th e spe c tru m o f Q.

N o tic e a lso th a t if, a t a po int v ∈ γ, (v) = 0 a s a n e le me nt o f k = R, it is

c le a rly re a so na b le to re d e fi ne δ a nd Q(v) b y d iv id ing b o th w ith (v). T h e n th e

en ergy d iff eren ces o f 1 / (v)Q(v) w ill c o me u p a s inte gra l v a lu e s.

U sing th e th e o re m (3 .3 ), w e fi nd th a t th e e v o lu tio n o pe ra to r U (τ0 , τ1 ) ma ps a ny

˜ ˜ ˜ ˜

ψi (τ0 ) to ψi (τ1 ) = e xp[ γ Q(τ )d τ ] (ψi (τ0 )) = e xp[ γ κi (τ )d τ ] (ψi (τ0 )). In pa rtic u la r

w e fi nd ,

˜

∂ ψi (τ ) ˜ ˜

= κi e xp( κi (τ )d τ )(ψi (τ0 )) = Q(ψi (τ )),

∂τ γ



o

so , o f c o u rse , a ga in th e S ch r¨ d inge r’s e q u a tio n, w ith τ a s time .

H e re o u r ” time -spa c e ” is S im pn (A(σ)), a nd S im p1 (A) is th e a na lo gu e o f th e

c la ssic a l c o nfi gu ra tio n spa c e . G iv e n a n e le me nt v ∈ S im pn (A(σ)), c o rre spo nd ing

to a simple mo d u le V o f d ime nsio n n, th e re a re fo r e v e ry a ∈ A(σ), a se t o f n

po ssib le v a lu e s, na me ly its e ige nv a lu e s, a s o pe ra to r o n V . S inc e V is simple , th e

stru c tu re ma p,

ρV : A(σ) −→ E nd k (V )

is su ppo se d su rje c tiv e , a nd so in ge ne ra l (a nd , fo r o rd e r 2 d yna mic a l syste ms,

˜ ˜

a lw a ys) th e o pe ra to rs ρ(a) a nd ρ(d a), a ∈ A, c a nno t a ll c o mmu te . In fa c t, if d im V =

∞, o r d im V ” a ppro a ch ing” ∞, se e e x a mple (3 .7 ), (3 .8 ), a ny o ne a ∈ A(σ) w o u ld

ρ ˜

te nd to h a v e a c o nju ga te , i.e . a n e le me nt b ∈ A(σ), su ch [˜(a), ρ(b)] = 1. T h e re fo re ,

˜ ˜

if th e v a lu e s qi o f ρ(a)) a re d e te rmine d , th e n th e v a lu e s pi o f ρ(b) w ill b e to ta lly

b ia se d , a nd v ic e v e rsa , giv ing u s th e H e ise nb e rg ind e te rmina c y pro b le m. In ge ne ra l

th e re is no w a y o f fi x ing a po int o f S im p1 (A(σ)) a s rep resen tin g V o r fi nd ing na tu ra l

mo rph isms,

S im pn (A) −→ S im pm (A(σ)), m m≥1 S im pm (A).



In th e v e ry spe c ia l c a se , w h e re A = k[x1 , ..., xp ] is a c o mmu ta tiv e po lyno mia l a l-

ge b ra , th e re e x ists mo re o v e r, fo r e v e ry line a r fo rm : V → k, a nd e v e ry sta te

˜

ψ(τ ) ∈ V |γ a c u rv e Ψ (γ) ⊂ S pe c(A) Ap d e fi ne d , b y its c o o rd ina te s, in th e

fo llo w ing w a y,

xi(τ ) = ˜

ρ(xi )ψ(τ )/ ψ(τ ), i = 1 , .., p.

GEOMETRY OF TIME-SPACES. 35



˜

Here V |γ is id en tifi ed w ith V ⊗k Oγ , τ b ein g a p a ra m eter o f γ. If w e a re a b le to p ick

˜ ˜ ˜

c o m m o n eig en fu n c tio n s {φj ∈ Vγ }, j = 1 , ..., n fo r ρ(xi ), i = 1 , ..., p, g en era tin g Vγ ,

i

w ith eig en v a lu es κj (τ ), j = 1 , ..., n, i = 1 , ..., p, a n d if ψ(τ ) = j λj (τ )φj , th en

p ick in g th e lin ea r fo rm d efi n ed b y , φj = 1 , j = 1 , ..., n, w e fi n d ,



xi (τ ) = λj (τ )κi (τ )/

j λj (τ ),

j j





w h ich is a g en era l fo rm o f E h ren fest’s th eo rem .

S u p p o se n o w th a t w e h a v e a situ a tio n w h ere th ere is a u n iq u e n o n -triv ia l p o sitiv e

(a s a rea l fu n c tio n ) P la n ck ’s c o n sta n t, ∈ C(n). C o n sid er f ∈ A(σ), a n d a ssu m e

ρ ˜

th a t th ere a re a m o n g th e {fλ }λ a c o n ju g a te, i.e. a fµ su ch th a t [˜(f ), ρ(f µ)] = 1.

T h is o b v io u sly c a n n o t h a p p en u n less d im V = ∞, b u t see th e ex a m p les (3 .7 ) a n d

(3 .8 ) fo r w h a t h a p p en s a t th e lim it w h en d im V g o es to ∞.

T h en w e ea sily fi n d th a t µ = − . M o reo v er, if ψ0 is a n eig en v ec to r fo r Q w ith

lea st en erg y (a ssu m ed a lw a y s p o sitiv e), κ0 , th en N := f− f is a quanta-counting

o p era to r, i.e. N (ψi ) = i, w h en κi = κ0 + (i − 1 ) , is th e i − th en erg y lev el. It

fo llo w s a lso th a t [Q, f− f ] = 0 . T h e a lg eb ra g en era ted b y {f , f− } is a k in d o f a

F ock re p re se ntation, F o n a F ock sp ace . Its L ie a lg eb ra o f d eriv a tio n s tu rn s o u t to

c o n ta in a V irasoro-lik e L ie-a lg eb ra . W e sh a ll retu rn to th is in th e ex a m p les (3 .7 )

a n d (3 .8 ) a t th e en d o f th is §, a n d in §5 .

W e h a v e seen th a t sta rtin g w ith a fi n itely g en era ted k-a lg eb ra A, a n d a d y n a m ic a l

sy stem σ, w e h a v e c rea ted a n in fi n ite series o f sp a c es S im pn (A(σ)) ⊂ S im p1 (C(n))

a n d a q u a n tu m fi eld th eo ry , d efi n ed o n th ese sp a c es, w ith tim e b ein g d efi n ed b y

th e D ira c d eriv a tio n .

˜

E a ch C(n) is c o m m u ta tiv e a n d V is a u n iv ersa l b u n d le o n S im p1 (C(n)), a n d

th e elem en ts o f A(σ), th e o b serv a b les, b ec o m e sec tio n s o f th e b u n d le o f o p era to rs,

E nd C(n) (V ). ˜

C lea rly , if D ⊂ S im p1 (C(n)) is a su b v a riety , sa y a c u rv e p a ra m etriz ed b y so m e

p a ra m eter q, th en th e u n iv ersa l fa m ily in d u c es a h o m o m o rp h ism o f a lg eb ra s,



ρD : A(σ) −→

˜ ˜

E nd D (V |D).



T h is is in m a n y rec en t tex ts referred to a s a quantifi cation o f th e c o m m u ta tiv e

a lg eb ra A(σ)/[A(σ), A(σ)], o r to a q u a n tu m d efo rm a tio n , a n d th e p a ra m eter q is

so m etim es c o n fo u n d ed w ith P la n ck ’s c o n sta n t. T h is is u n fo rtu n a te, b u t p ro b a b ly

u n a v o id a b le!

In q u a n tu m th eo ry o n e a ttem p ts to trea t th e second quantifi cation o f a n o sc illa to r

in d im en sio n 1 , a s a c erta in rep resen ta tio n o n th e F o ck sp a c e, i.e. c o n stru c tin g

o b serv a b les a c tin g o n F o ck sp a c e, w ith th e p ro p erties o n e w a n ts. T h is tu rn s o u t

to b e rela ted to th e c a n o n ic a l rep resen ta tio n s o f o u r P h C := k o n a n

n-b u n d le o v er th e a lg eb ra , R := k[[n]p ,q ]. Here th e p, q-d e form ed num be rs [n]p ,q a re

in tro d u c ed a s,



[n]p ,q := q n−1 + pq n−2 + p2 q n−3 + ... + pn−2 q + pn−1 ,



a n d w e m a y a s w ell c o n sid er p, q a s fo rm a l v a ria b les, so th a t R ⊂ k[p, q]. O n e o b ta in s

a h o m o m o rp h ism o f A(σ) in to a n en d o m o rp h ism rin g o f th e fo rm E nd R (V ⊗k R),

see e.g . ([E lb a z ], A p p en d ic e, o n th e ” q -c o m m u ta to rs” ). P ick in g rep resen ta tiv es

36 OLAV ARNFINN LAUDAL



for x a n d dx in Mn (R), it tu rn s ou t th a t, in ste a d of g e ttin g th e c la ssic a l d e fi n in g

re la tion s for a n osc illa tor, i.e . a H a m ilton ia n Q, su ch th a t in E ndR (V ⊗k R),



a+ := x + dx, a− : x − dx, [Q, x] = dx, [Q, dx] = x, [a− , a+ ] = 1



on e fi n d s,



a+ := x + dx, a− : x − dx, [Q, x]q = dx, [Q, dx]q = x, [a− , a+ ]q = 1



w h e re [a, b]q := ab − qba is th e ” q u a n tiz e d ” c om m u ta tor. T h is h old s in p a rtic u la r

for p = 1 , so for R = k[q], d e fi n in g a c u rv e D in S im pn (P h k[x]).

H ow e v e r, th is k[q]-p a ra m e triz a tion is n ot p a ra m e triz in g a n in te g ra l c u rv e of ξ in

S im pn (P h C). O n th e c on tra ry , it is p a ra m e triz in g a c u rv e w h ich is, tra n sv e rsa l to

ξ, a n d th e re fore re p re se n t a p h e n om e n on w h ich ta k e s p la c e in sta n ta n e ou sly , se e th e

e x a m p le s (3 .7 ), (3 .8 ).

Example 3.4. L e t C b e a fi n ite ty p e c om m u ta tiv e k-a lg e b ra , sa y p a ra m e triz in g

a n in te re stin g m od u li sp a c e , a n d a ssu m e it is n on -sin g u la r, a n d p ick a sy ste m of

re g u la r c oord in a te s {t1 , t2 , ..., tr } in C. L e t L ∈ P h C b e a L a g ra n g ia n , i.e . a n

e le m e n t d e fi n in g a n ” a c tion ” in S im pn (P h C), th e n th e E u le r-L a g ra n g e d iff e re n tia l

e q u a tion s

∂L ∂L

δ( )− = 0 , for a ll 1 ≤ l

∂dtl ∂tl

d e te rm in e s a d y n a m ic a l sy ste m σ. S u p p ose it is of ord e r 2 . C on sid e r,



C(σ) := P h ∞ C/I (n) P h C/(σ)



w ith D ira c d e riv a tion δ, d e te rm in e d b y (σ).

W e c ou ld sta rt w ith L := g = i,j= 1,..,r gi,j dti dtj ∈ P h C, a R ie m a n n ia n m e tric

d e fi n e d on S im p1 (C). It is e a sy to c om p u te th e E u le r-L a g ra n g e e q u a tion s. W e fi n d

th e force laws,

¯

d2 tl = −Γl := − ¯ i,j

Γl dti dtj

¯ ¯

w h e re Γl := 1 /2 (Γl + Γl ) a re th e C h ristoff e l sy m b ols for th e L e v i-C iv ita c on -

i,j i,j j,i

n e c tion of g. T h is is c le a rly a d y n a m ic a l sy ste m σ := σ(g) of ord e r 2 , a n d so,



C(σ(g)) := P h C/(σ)



a s k-a lg e b ra . S in c e in C(σ) th e D ira c d e riv a tion h a s th e form ,



∂ ∂

δ= (dtl + Γl ),

∂tl ∂dtl

l





th e c orre sp on d in g fu n d a m e n ta l v e c tor fi e ld ξ in S im p1 (C(σ)) = S pe c(k[ti , uj ], uj :=

dtj ), is,

∂ ∂

ξ= (ul + Γl )

∂tl ∂ul

l



Its in te g ra l c u rv e s p roje c ts on to S im p1 (C) to g iv e th e g e od e sic s of th e m e tric g.

GEOMETRY OF TIME-SPACES. 37



Now consider the Levi-Civita-connection,



: θH −→ E ndk (θH )



ex p ressed in coordinates as,



∂ ∂

∂ ( ∂t j ) = Γl

i,j

∂t l

l

∂t i

Classsically we defi ne the cu rvatu re of a sp ace, as the ob stru ction for to b e a

∂

Lie-alg eb ra hom om orp hism . P u t δi = , then we fi nd,

∂t i

l

[ δi , δj ](δk ) = Ri,j,k δl .

l



T his, ob viou sly , is a com m u tative version of the m ore p recise cu rvatu re



¯

d3 tl = Rl (t, dt).



S ince in P h H the dtj ’s and the ti ’s do not com m u te, we cannot, in g eneral, write



d 3 tl = ¯l

Ri,j,k δi δj δk .

i,j,k



R ecall that the R icci tensor is g iven as,

j

Ric i,k = Ri,j,k

j



and that,

R := g k,i Ric i,k

k,i



is the scalar cu rvatu re of g, som etim es called S.

S ee [La 5 ]. for the sp ecial case of ou r ” toy ” m odel, the m odu li of of an ob server-

ob served in E u clidean 3 -sp ace.

Example 3.5. Let u s g o b ack to the case of A = k , the free non-

com m u tative k-alg eb ra on two sy m b ols, and the dim ension n = 2 , see (2 .1 4 ). W e

fou nd,

C(2 ) k[t1 , t2 , t3 , t4 , t5 ].

˜

locally in a Z arisk i neig hb orhood of the orig in. T he versal fam ily V , is defi ned b y

the actions of x1 , x2 , g iven b y ,



0 1 + t3 t1 t2

X1 := , X2 := .

t5 t4 1 + t3 0



T he F orm anek center, in this case. is cu t ou t b y the sing le eq u ation:



f := de t[X1 , X2 ] = −((1 + t3 )2 − t2 t5 )2 + (t1 (1 + t3 ) + t2 t4 )(t4 (1 + t3 ) + t1 t5 ).

38 OLAV ARNFINN LAUDAL



and



trX1 = t4 , trX2 = t1 ,

de tX1 = −t5 − t3 t5 , de tX2 = −t2 − t2 t3 ,

tr(X1 X2 ) = (1 + t3 )2 + t2 t5 ,



so th e trace ring o f th is fam ily is



k[t1 , t2 + t2 t3 , 1 + 2t3 + t2 + t2 t5 , t4 , t5 + t3 t5 ] =: k[u1 , u2 , u3 , u4 , u5 ],

3



w ith ,



u1 = t1 , u2 = (1 + t3 )t2 , u3 = (1 + t3 )2 + t2 t5 , u4 = t4 , u5 = (1 + t3 )t5 ,



and f = −u2 + 4u2 u5 + u1 u3 u4 + u2 u5 + u2 u2 . M o re o v e r, k[t] is alg e b raic o v e r k[u],

3 1 4

w ith disc rim inant, ∆ := 4u2 u5 (u2 − 4u2 u5 ) = 4(1 + t3 )2 t2 t5 ((1 + t3 )2 − t2 t5 )2 , and

3

e

th e re is an ´ tale c o v e ring ,



A5 − V (∆ ) → S im p 2 (A) − V (∆ ).



N o tic e th at if w e p u t t1 = t4 = 0 , th e n f div ide s ∆ .

W ith th is do ne , le t u s c o nside r q u antu m th e o ry in dim e nsio n 1. T h at is, w e start

w ith th e k-alg e b ra C = k = k[x], and c o nside r th e c lassic al L ag rang ians,



L = 1/2dx2 + V (x) ∈ P h C.



T h e c o rre sp o nding dy nam ic al sy ste m σ is g iv e n b y th e e q u atio n,



∂V

d2 x = ,

∂x

and is o f o rde r 2, so th e alg e b ra o f inte re st is,



C(σ) = P h C = k k



L e t u s fi rst c o m p u te th e p articles o f rank 1 fo r so m e c ase s, and le t u s start w ith

V (x) = ±1/2 x2 , i.e . th e c lassic al o sc illato r. T h e fu ndam e ntal e q u atio n o f th e

dy nam ic al sy ste m is,

δ = [δ] + [Q, −],

w h e re , in dim e nsio n 1, th e e ndo m o rp h ism Q o b v io u sly c o m m u te s w ith th e ac tio ns

o f xi , i = 1, 2. T o so lv e th e e q u atio n ab o v e , w e m ay th e re fo re fo rg e t ab o u t Q, so

w e are le ft w ith th e v e c to r fi e lds,

[δ] = ξ .

T h e sp ac e , S im p 1 (C(σi )), is ju st th e o rdinary p h ase sp ac e , S im p 1 (k[x, dx]). P u t

as ab o v e , x1 := x, x2 := dx. W e m u st th e re fo re ju st so lv e th e e q u atio ns,



δ(x) =[δ](x) = [δ](x1 )

δ 2 (x) =[δ](dx) = [δ](x2 )

GEOMETRY OF TIME-SPACES. 39



We can obviously pick,

∂

δi = χ i = ,

∂x i

so w e m ust h ave

∂ ∂

[δ] = ξ1 + ξ2 .

∂x 1 ∂x 2

In th e case of th e potential, V = 1/2x2 , w e g et th e eq uations,



x2 =[δ](x) = [δ](x1 ) = ξ1

x1 =[δ](d x) = [δ](x2 ) = ξ2



T h erefore th e fund am ental vector fi eld is,



∂ ∂

ξ = x2 + x1

∂x 1 ∂x 2



i.e. w e fi nd h yperbolic m otions in th e ph ase space, w ith g eneral solutions,



x = x1 = r cosh (t + c), d x = x2 = r sinh (t + c)



w h ich is w h at w e ex pected .

In th e case of th e oscillator, V = −1/2x2 , w e g et th e eq uations,



x2 =[δ](x) = [δ](x1 ) = ξ1

−x1 =[δ](d x) = [δ](x2 ) = ξ2



T h erefore th e fund am ental vector fi eld is,



∂ ∂

ξ = x2 − x1

∂x 1 ∂x 2



i.e. w e fi nd circular m otions in th e ph ase space, w ith g eneral solutions,



γ : x = x1 = r cos(t + c), d x = x2 = −r sin(t + c),



w h ich is also w h at w e ex pected . C onsid er now th e versal fam ily restricted to γ,



˜ ˜

ργ : k → E n d γ (V |γ),



˜

and a state ψ(t) ∈ V |γ. If Q, restricted to γ, is m ultiplication by κ(t), th en th e

o

S ch r¨ d ing er eq uation becom es,



∂

ψ = κ(t)ψ

∂t



so th at w e sh ould h ave,

t

ψ(t) = e xp ( κ)!

γ

40 OLAV ARNFINN LAUDAL



˜

This will turn out much nicer, if we extend the action of k to VC , and

p ut Q, restricted to γ, eq ual to multip lication b y ıκ(t). Then we fi nd the reasonab le

result,

t

ψ(t) = e xp (ı κ).

γ



S ee ag ain [L a 5].

In the rep ulsiv e, resp . attractiv e, N ewtonian case, with V = ±1/x, we fi nd,





x2 =[δ](x) = [δ](x1 ) = ξ1

(1/x2 ) =[δ](d x) = [δ](x2 ) = ξ2 ,

1 = +, −.



Therefore the fundamental v ector fi eld is,



∂ ∂

ξ = x2 + (1/x2 )

1

∂x 1 ∂x 2



with the classical solution,

x = (9 /2)t2/3 .



Example 3.6. F or the oscillator, in rank 2, thing s are more diffi cult. A s we hav e

comp uted ab ov e, we hav e found a (p artial) v ersal family of S im p 2 (P h k[x]), g iv en

b y,

0 1 + t3 t1 t2

x= ,dx =

t5 t4 1 + t3 0

and we may chose,

∂

χi = , i = 1, 2, ..., 5.

∂ti

and, ob v iously ,

∂

δi = , i = 1, 2, ..., 5.

∂ti

The fundamental v ector fi elds will hav e the form,



∂

[δ] = ξi δi , ξ = ξi ,

∂ti



with 5 unk nowns, ξi , i = 1, 2, .., 5 M oreov er,



q1,1 q1,2

Q= ,

q2,1 q2,2



with 4 unk nowns qi,j , i = 1, 2, j = 1, 2. N ow, recall that Q can only b e determined

up to a z entral element from M2 (C), i.e. we hav e 8 essential unk nowns, ξi , i =

1, 2, 3, 4, 5 and (q1,1 − q2,2 ), q1,2 , q2,1 in the two matrix eq uations,



δ(x) = d x = [δ](x) + [Q, x]

δ 2 (x) = ±x = [δ](d x) + [Q, d x]

GEOMETRY OF TIME-SPACES. 41



On the right hand side of the equations we have the terms,



0 1 + t3 0 ξ3

[δ](x) = ξi δi ( )=

t5 t4 ξ5 ξ4

t1 t2 ξ1 ξ2

[δ](d x) = ξi δi ( )=

1 + t3 0 ξ3 0



and the terms,



t5 q1,2 − (1 + t3 )q2,1 (1 + t3 )q1,1 + t4 q1,2 − (1 + t3 )q2,2

[Q , x] =

t5 q2,2 − t5 q1,1 − t4 q2,1 (1 + t3 )q2,1 − t5 q1,2

(1 + t3 )q1,2 − t2 q2,1 t2 q1,1 − t1 q1,2 − t2 q2,2

[Q , d x] = ,

t1 q2,1 + (1 + t3 )q2,2 − (1 + t3 )q1,1 t2 q2,1 − (1 + t3 )q1,2



and on the left side, we have,



t1 t2

δ(x) = d x =

1 + t3 0

0 1 + t3

δ 2 (x) = ±x = ±

t5 t4



W riting up the matrix for the c orresp onding linear equation, we fi nd that the

determinant of the 8 × 8 matrix turns out to b e easily c omp uted, it is,



D = 2(1 + t3 )(t2 t5 − (1 + t3 )2 ).



N otic e that D is a divisor in the disc riminant, ∆ = 4 (1 + t3 )2 t2 t5 ((1 + t3 )2 − t2 t5 )2 ,

see (3 .5 ). M oreover we fi nd,



(q1,1 − q2,2 )+ = D−1 (−(1 + t3 )(t2 + t2 ) + (t2 − t5 )(t2 t5 − (1 + t3 )2 − t1 t4 ))

1 4



(q1,1 − q2,2 )− = D−1 ((1 + t3 )(t2 − t2 ) + (t2 − t5 )(t2 t5 − (1 + t3 )2 − t1 t4 ))

1 4

− t1 t4 (1 + t3 ))

q1,2 = D−1 (2(1 + t3 )(t1 t2 ± (1 + t3 )t4 )

q2,1 = D−1 (2(1 + t3 )(t4 t5 ± t1 (1 + t3 ))

ξ1 = t2 q2,1 − (1 + t3 )q1,2

ξ2 = −t2 (q1,1 − q2,2 ) + t1 q1,2 ± (1 + t3 )

ξ3 = (1 + t3 )(q1,1 − q2,2 ) + t1 q2,1 ± t5

ξ4 = t5 q1,2 − (1 + t3 )q2,1

ξ5 = t5 (q1,1 − q2,2 ) + t4 q2,1 + (1 + t3 )



P ick , V (x) = +1 /2x2 , and see that ξ1 = ξ4 = 0 imp ly ,



((1 + t3 )2 − t2 t5 )q1,2 = ((1 + t3 )2 − t2 t5 )q2,1 = 0,



and therefore, q1,2 = q2,1 = 0. S inc e we assume that ∆ = 0, this also imp lies that

t1 = t4 = 0. T herefore the singularities of ξ are given, b y ,



t2 = −/ + (1 + t3 ), t5 = +/ − (1 + t3 ),

42 OLAV ARNFINN LAUDAL



or, uniquely, by the representation,



0 1

x=

1 0

0 −1

dx =

1 0

q1,1 0

Q= .

0 q1,1 + 1



c orrespond ing to t1 = 0, t2 = −1, t3 = 0, t4 = 0, t5 = 1. N otic e that in this c ase

(i.e. w hen V (x) = 1/2x2 ), w e fi nd , in all d im ensions, that f := ρ(x + d x), is an

eig env ec tor for [Q, −] w ith f− = ρ(x − d x) so that N = f− f is the quantum

c ounting operator.

N ow , to fi nd the integ ral c urv es of the v ec tor fi eld ξ, w e m ust solv e the obv ious

∂ ti

system of d iff erential equations, = ξi , i = 1, .., 5 . It turns out that w e are m ostly

∂τ

interested in the solutions for w hich there ex ists sing ular point, c orrespond ing to

t1 = t4 = 0. If they ex ist they look lik e,



∂ t1

= ξ1 = 0

∂τ

∂ t2

= ξ2 = −t2 (t2 − t5 )(2 + 2t3 )−1 ± (1 + t3 )

∂τ

∂ t3

= ξ3 = 1/2(t2 − t5 ) ± t5

∂τ

∂ t4

= ξ4 = 0

∂τ

∂ t5

= ξ5 = t5 (t2 − t5 )(2 + 2t3 )−1 + (1 + t3 ).

∂τ

A nd these equations are obv iously c onsistent w ith the c ond itions t1 = t4 = 0.

M oreov er, introd uc ing new v ariables,



y1 = (t2 − t5 )

y2 = (t2 + t5 )

y3 = (2 + 2t3 )



so that,



t2 = 1/2 (y2 + y1 )

t5 = 1/2 (y2 − y1 )

t3 = 1/2 y3 − 1.



thing s look nic er. W e fi nd ,



ξ2 = − 1/2 (y1 + y2 ) + 1/2 y3

ξ3 =1/2 y2

ξ5 =1/2 (y2 − y1 )y1 y3

−1

GEOMETRY OF TIME-SPACES. 43



In the new coordinates the system of equations above reduces to,

∂y1 ∂y2 ∂y3

y1 − y2 + y3 =0

∂τ ∂τ ∂τ

−1 ∂y1 −1 ∂y3

y1 + y3 = 0.

∂τ ∂τ

T he integ ral curves are therefore intersections of the form,

2 2 2

C(c1 , c2 ) := V (y1 − y2 + y3 = c1 ) ∩ V (y1 y3 = c2 ).

M oreover, the stratum at infi nity, g iven by f = 0, where f is the F ormanek center,

is now easily comp uted, in terms of the new coordinates it is g iven as,

2 2 2

f = −1/16(y1 − y2 + y3 )2

T his shows that a p article corresp onding to an integ ral curve γ := C(c1 , c2 ), with

c1 = 0 lives eternally, as it should. Its comp letion does not intersect the F ormanek

center, i.e. the stratum at infi nity.

A n easy calculation g ives us, see (3.5 ),

2 2 2

16(y1 − y2 + y3 )2 = −u2 + 4u2 u5 ,

3

y1 y3 = 2(u2 − u5 )

so the integ ral curve of the harmonic oscillator will be p lane conic curves in the

p art of S im p 2 (P h k[x]), where ∆ = 0, u1 = u4 = 0, g iven by,

u2 − 4u2 u5 = c3 , (u2 − u5 ) = c4 .

3



H ere c3 = 0, c4 are constants. N otice also that our sp ecial p oint, the sing ularity for

ξ, g iven by y1 = −2, y2 = 0, y3 = 2, sits on the curve defi ned by c1 = 8 , c2 = −4,

corresp onding to c3 = 32, c4 = −2.

In the new, y-coordinates, the versal family of S im p 2 (P h k[x]), lifted to U (2),

and restricted to t1 = t2 = 0, is g iven by,

0 1/2y3 0 1/2(y1 + y2 )

x= ,dx = .

1/2(y2 − y1 ) 0 1/2y3 0

M oreover along the curve γ, defi ned by c1 = 8 , c2 = −4, which is g iven by the

equations,

2 2 2

y3 = −4y1 , y2 = y1 + 16y1 − 8 = (y1 − 4)2 y1 ,

−1 −2 −2





the vectorfi eld ξ is g iven by,

∂

ξ = −1/4(y1 + 2)(y1 − 2)y1

∂y 1

or,

∂

ξ = 3/4(y1 + 2)(y1 − 2)y1 ,

∂y 1

dep ending on which root we choose for y2 above. T he corresp onding tim e along

γ, is then g iven as, τ = −lo g (y1 ) + 1/2lo g (y1 + 2) + 1/2lo g (y1 − 2), resp ectively

τ = 1/3lo g (y1 ) − 1/6lo g (y1 + 2) − 1/6lo g (y1 − 2), both with a sing ularity at y1 =

−2, y1 = 2, corresp onding to the same unique sing ularity of ξ, in S im p 2 (P h (k[x]).

T he versal family is not defi ned at y1 = 0, see above.

44 OLAV ARNFINN LAUDAL



Example 3.7. (i) We shall not treat oscillators in rank≥ 3 , in g eneral, b u t only

look at the sing u larities, in all ranks. T his is all w ell know n in p hy sics, see [E lb az ],

section 1 6 , althou g h in m ost b ooks in p hy sics, it is treated rather form ally , in

relation w ith the second quantification and the introd u ction of F ock-sp aces, and

their associated rep resentations of the alg eb ra of ob serv ab les. We shall see that

this second q u antifi cation is a natu ral q u otient of the alg eb ra of ob serv ab les P hC,

in line w ith the g eneral p hilosop hy of this p ap er. A lthou g h w e m ay w ork in a v ery

g eneral setting , w e shall, as ab ov e, restrict ou r attention to the classical oscillator

(V (x) = 1 /2x2 ), in d im ension 1 .

A s ab ov e w e fi nd ,

d2 x = x

and the D irac d eriv ation has therefore,

a+ := 1 /2(x + dx), a− := 1 /2(x − dx)

as eig env ectors, w ith eig env alu es 1 and -1 resp ectiv ely . S ince P hC = k

is g enerated b y the elem ents a+ := 1 /2(x + dx), a− := 1 /2(x − dx), it is clear that

P lanck’s constant = 1 .

U sing the m ethod e ab ov e it is easy to see that for any rank n = dimV , a

sing u lar p oint v ∈ S imp n (P hC) corresp ond s to a k -m od u le V , w ith

x and dx acting as end om orp hism s X, dX ∈ E ndk (V ) for w hich there ex ists an

end om orp hism , the H am iltonian, Q ∈ E ndk (V ) w ith,

dX := ρ(dx) = [Q, ρ(x)] =: [Q, X]

X = ρ(d2 x) = [Q, ρ(dx)] =: [Q, dX]

L et ψ0 b e any eig env ector for Q w ith eig env alu e κ0 . S ince V is sim p le, the fam ily

{am an (ψ0 )} m u st g enerate V . M oreov er, if am an (ψ0 ) = 0, w e know it m u st b e

+ − + −

an eig env ector for Q, w ith eig env alu e κ0 + (m − n). We can, b y ad d ing λ1 to Q,

assu m e that there is a b asis for V of eig env ectors for Q, w ith eig env alu es of this

form . T his m eans that Q can b e assu m ed to hav e the form ,

κ0 0 0 0 ... 0

 

 0 κ0 + λ 1 0 0 ... 0 

Q= 0 0 κ0 + λ2 0 ... 0 ,

 

. . . . ... 0

 

0 0 0 ... 0 κ0 + λn−1

w here 0 ≤ λ1 ≤ λ2 ≤ ... ≤ λn−1 are all integ ers. M oreov er, since V is sim p le, and

[Q, a+ ] = a+ , [Q, a− ] = −a− , an easy com p u tation show s that,

0 0 0 0 ... 0

 

 a2,1 0 0 0 ... 0

a+ =  0 a3,2 0 0 ... 0,

 

. . . . ... 0

 

0 0 0 ... an,n−1 0

0 a1,2 0 0 ... 0

 

0 0 a2,3 0 ... 0 

a− =  0 0 0 0 ... 0 ,

 

. . . . ... an−1,n

 

0 0 0 ... 0 0

GEOMETRY OF TIME-SPACES. 45



where all ai,i−1 , ai,i+1 = 0. W e also fi n d ,

[a+ , a− ]

−a1,2 a2,1 0 0 0 ... 0

 

 0 a2,1 a1,2 − a2,3 a3,2 0 0 ... 0 

= 0 0 a3,2 a2,3 − a3,4 a4,3 0 ... 0

 



. . . . ... 0

 

0 0 0 ... 0 an,n−1 an−1,n

o b v io u sly with v an ishin g trac e.

N o w to hav e the c lassic al fo rm u las, see ([E lb az ], p .37 7 -38 0), we ju st hav e to

im p o se the c o n d itio n that a+ an d a− b e conjugate o p erato rs, i.e. that

−1 0 0 ... 0

 

 0 −1 0 ... 0 

[a+ , a− ] =  0 0 −1 ... 0 .

 

. . . .... 0

 

0 0 0 ... (n − 1)

T hen , in tro d u c in g a b ase chan g e, c o rresp o n d in g to an in n er au to m o rp hism d efi n ed

b y a d iag o n al m atrix , we fi n d that we m ay assu m e ai,i+1 = ai+1,i . It fo llo ws that,

 √ 

√0 1 √0 0 ... 0

 1 0 2 √ 0 ... 0

√



 

X= 0  2 0 3 ... 0 



 . . . . ... (n − 1) 

0 0 0 ... (n − 1) 0

 √ 

√0 − 1 0

√ 0 ... 0

 1 0 − 2 0 ... 0

√ √



 

dX =   0 2 0 − 3 ... 0 



 . . . . ... − (n − 1) 

0 0 0 ... (n − 1) 0

with asso c iated H am ilto n ian ,

1/2 0 0 0 ... 0

 

 0 3/2 0 0 ... 0 

Q= 0 0 5 /2 0 ... 0 .

 

. . . . ... 0

 

0 0 0 ... 0 (2n − 1)/2

C learly , we c an n o t im p o se, [a− , a+ ] = 1, in fi n ite ran k . If, ho wev er, we let n =

d im V ten d to ∞, then we fi n d ex ac tely the c lassic al fo rm u las fo r the o sc illato r in

the ” sec o n d q u an tifi c atio n ” , see the referen c e ab o v e. In p artic u lar it fo llo ws that

[a− , a+ ] = 1 is the o n ly relatio n b etween the o p erato rs a− an d a+ in this c lassic al

lim it rep resen tatio n .

W e m ig ht try to fi n d fu n c tio n s, o r fo rm al p o wer series, [n] ∈ k[[τ ]] su ch that the

rep resen tatio n ,

0 [1] 0 0 ... 0

 

 [1] 0 [2] 0 ... 0 

 

x(n) =   0 [2] 0 [3] ... 0 



 . . . . ... [(n − 1)] 

0 0 0 ... [(n − 1)] 0

46 OLAV ARNFINN LAUDAL









0 − [1] 0 0 ... 0

 



 [1] 0 − [2 ] 0 ... 0 



dx(n) = 

 0 [2 ] 0 − [3 ] ... 0 



 . . . . ... − [(n − 1)] 

0 0 0 ... [(n − 1)] 0

w ith a sso c ia te d H a m ilto n ia n ,

1/2 + [0] 0 0 0 ... 0

 

 0 1/2 + [1] 0 0 ... 0 

Q= 0 0 1/2 + [2 ] 0 ... 0

 



. . . . ... 0

 

0 0 0 ... 0 1/2 + [n − 1]

sa tisfi y th e fu n d a m e n ta l d y n a m ic a l e q u a tio n ,



δ = [δ] + [Q, −].

∂

W e m a y , o f c o u rse ch o o se [δ] := a s th e g e n e ra to r o f th e v e c to r fi e ld s o n th e

∂τ

τ -lin e . W e fi n d th e fo llo w in g sy ste m o f d iff e re n tia l e q u a tio n s,

∂ 2 2

fn + (fn − fn−1 )fn = fn

∂τ

∂ 2

fn + (−fn − f + n − 12 )fn = −fn

∂τ

w h e re fn := [n], a n d w ith b o u n d a ry c o n d itio n s,



fn (0)2 = n.



T h e se e q u a tio n s im m e d ia te ly le a d to c o n sta n t fn s, a n d th e re fo re p ro v e s th a t

th e c u rv e in S im p n (P h C ) d e fi n e d b y th e fa m ily {x(n), dx(n)} is tra n sv e rsa l to th e

fu n d a m e n ta l v e c to r fi e ld ξ. T h e in tro d u c tio n o f th e (p ,q ) c o m m u ta to rs, a n d th e ir

tre a tm e n t in p h y sic s, m a k e s it p o ssib le to tre a t th e fe rm io n s a n d th e b o so n s in a

c o m m o n stru c tu re . L e ttin g th e p a ra m e te r q in th e a b o v e fa m ily slid e fro m 1 to

-1, th e q-c o m m u ta to r [−, −]q ch a n g e s fro m th e o rd in a ry L ie p ro d u c t to th e J o rd a n

p ro d u c t. T h e c o m p u ta tio n a b o v e sh o w s th a t th is ch a n g e ta k e s p la c e tra n sv e rsa l to

tim e , i.e . in sta n ta n o u sly !

(ii) F o r th e h a rm o n ic o sc illa to r in d im e n sio n n ≥ 2 w e h a v e A = k[x1 , x2 ], a n d ,

P h A = k /([x1 , x2 ], [x1 , dx2 ] − [x2 , dx1 ]), a n d ,



A(σ) = k /([x1 , x2 ], [x1 , dx2 ] − [x2 , dx1 ], [dx1 , dx2 ]).



M o re o v e r, in ra n k 2 w e fi n d a sim p le re p re se n ta tio n o f A(σ), g iv e n b y ,



1 0 0 0

X1 = , X2 =

0 0 0 1

0 −1 0 1

dX1 = , dX2 =

1 0 −1 0

GEOMETRY OF TIME-SPACES. 47



with,

0 −1

[X1 , dX1 ] = [X2 , dX2 ] = .

−1 0



T he p ro b le m o f in te g ra tin g the d iff e re n tia l e q u a tio n s a b o v e , i.e . fi n d in g a lg e b ra ic

g e o m e tric fo rm u la s fo r the in te g ra l c u rv e s o f ξ, is a c la ssic a l p ro b le m , a n d we m a y

u se a te chn iq u e a lre a d y we ll k n o wn to H a m ilto n a n d J a c o b i. In fa c t, a ssu m in g tha t

A = k[t1 , ...,n ], a n d tha t σ is d e te rm in e d b y the fo llo win g force-laws,



d2 ti = Γ i (t1 , ..., tn , dt1 , ..., dtn )



we ha v e tha t,

n

∂ ∂

A(σ) = P h ∞ (A)/(σ), δ = (dti + Γi ).

i= 1

∂ti ∂dti



W e m a y try to so lv e the e q u a tio n ,



δθ = 0



in the rin g A(σ). O b v io u sly the se t o f so lu tio n s fo rm a su b -rin g o f A(σ), the rin g

o f in v a ria n ts, a n d we ha v e the fo llo win g e a sy re su lt,

P ro p o sitio n 3 .8 . L et Θ = ke r δ, be th e su b rin g of in v arian ts in A(σ).

(i) S u p p ose ρ : A(σ) → E ndk (V ) is a rep resen tation for wh ich th e tan g en t sp ace

of V , E x t1 (V, V ) = 0, or su p p ose V corresp on d s to a p oin t t ∈ S im p n (A(σ))

A(σ)

for wh ich ξ(t) = 0, th en an y θ ∈ Θ is con stan t in V , i.e. [Q, ρ(θ)] = 0, so th at th e

eig en v ectors of Q are eig en v ectors for θ.

(ii) C on sid er for an y n th e u n iv ersal fam ily ,



˜ ˜

ρ : A(σ) → E ndC(n) (V ).



an d let θ ∈ Θ, th en

˜

tr a ce ρ(θ) ∈ C(n)

is con stan t alon g an y in teg ral cu rv e of ξ in S im p n (A(σ)).

P roof. (i) S u p p o se δ(θ) = 0, a n d c o n sid e r the d y n a m ic a l e q u a tio n ,



δ = [δ] + [Q, −],



whe re we m a y a ssu m e [δ] = ξ. If the ta n g e n t sp a c e o f V is triv ia l the n , o b v io u sly

[δ] = 0 the re fo re δ(θ) = 0 im p lie s [Q, ρ(θ)] = 0.

˜

(ii) If δ(θ) = 0, we m u st ha v e , in E ndC(n) (V ),



ρ ˜ ρ ˜

0 = tr a ce ξ(˜(θ)) + tr a ce [Q, ρ(θ)] = tr a ce ξ(˜(θ)) = ξ(tr a ce ρ(θ)).







O n the b a sis o f the e x a m p le s a b o v e , in p a rtic u la r (3 .6 ), it is te m p tin g to c o n je c -

tu re tha t a ll in te g ra l c u rv e s o f ξ a re in te rse c tio n s o f hy p e rsu rfa c e s o f S p e c(C(n)),

48 OLAV ARNFINN LAUDAL



ρ

of the form traceξ(˜(θ)) = 0. H ow ev er, this is n ot tru e, a s w e c a n see b y g oin g b a ck

to (3 .6 ). H ere w e ha v e



∂ ∂

A = k[x], A(σ) = P h A = k = k , y = d x, δ = y +x .

∂x ∂y



T here a re on ly tw o ob v iou s in v a ria n ts, θ1 = x2 − y 2 a n d θ2 = xy − yx. M oreov er

the u n iv ersa l fa mily on C(2) = k[t1 , .., t5 ], is g iv en b y ,



0 1 + t3 t1 t2

˜

ρ(x) = ˜

, ρ(y) = .

t5 t4 1 + t3 0



W e fi n d , see (3 .5 ), tha t the in v a ria n ts ex p ressed in the c oord in a tes (u1 , ..., u5 ), look s

lik e,



ρ

trace(˜(θ1 )) = − u1 − 2u2 + u4 + 2u5

d et(˜(θ1 )) =(u5 − u2 − u2 )(u5 − u2 + u2 ) − u2 u5 + u1 u3 u4 − u2 u2 .

ρ 1 4 4 1



d et(˜(θ2 )) = − u2 + 4u2 u5 + u1 u3 u4 + u2 u5 + u2 u2

ρ 3 1 4

ρ ρ

d et(˜(θ1 )˜(θ2 )) =0.



ρ

If w e p u t t1 = t4 = 0, w e fi n d the resu lt of (3 .6 ), n a mely trace(˜(θ1 )) = y1 y3 =

2 2 2

2(u5 −u2 ), d et(˜(θ1 )) = 1/4(y1 y3 ) = (u5 −u2 )2 , d et(˜(θ2 )) = −1/16 (y1 −y2 +y3 )2 =

ρ ρ 2 2

2

ρ ρ

−u3 + 4u2 u5 . H ow ev er, the fa c t tha t d et(˜(θ1 )˜(θ2 )) = 0 in d ic a tes tha t there

a re n on -a lg eb ra ic in teg ra l c u rv es sittin g on a n a lg eb ra ic su rfa c e of A5 . N otic e

a lso tha t the non-commutative invariant θ2 is essen tia l in the in teg ra tion of ξ in

this c a se. N otic e a lso tha t in the c a se A = k[x1 , x2 , x3 ], if the L a g ra n g ia n L =

∂U ∂U

1/2(d x2 + 1/2(d x2 + 1/2(d x2 ) + U , ha s a p oten tia l U , su ch tha t

1 2 3 xj = xi ,

∂x i ∂x j

i.e. c on c ern s a central force then the a n g u la r momen ta Li,j := xi d xj − xj d xi , a re

c on sta n ts, i.e. δ(Li,j ) = 0, w hich of c ou rse ha v e the c la ssic a l c on seq u en c es on e

k n ow s. C omb in in g this w ith the rep resen ta tion s d isc u ssed in the E x a mp le(1.1),

(iii), w e fi n d in terestin g resu lts, see a forthc omin g p a p er on this su b jec t.

E x a m p le 3 .9 . F or the q u a rtic anh armonic osc illa tor, g iv en b y L = 1/2 d x2 +

1/4 αx4 w e ma y ea sily c omp u te the ra n k 2 a n d 3 v ersa l fa milies. In ra n k 2 w e

fi n d tha t there is a on e-d imen sion a l sin g u la r fa mily of d imen sion 2 simp le mod u les,

w ith,

0 αt3 0 −α2 t5 0 0

X= ,dX = ,Q = X = .

t 0 αt3 0 0 αt2

In ra n k 3 w e fi n d tha t there a re n o simp le sin g u la r mod u le w ith c orresp on d in g

d ia g on a l H a milton ia n . T his ma y b e on e rea son w hy the en erg y lev els of the q u a rtic

a n ha rmon ic osc illa tor is n ot k n ow n to the p hy sic ists.

E x a m p le 3 .1 0 . N ow , let u s c on sid er the in fi n ite ra n k c a se. In p a rtic u la r w e ma y

c on sid er the rep resen ta tion g iv en in the a b ov e ex a mp le, w hen n = d im V ” b ec omes”

∞. N otic e tha t this is g iv en a s the limit c a se of the sin g u la r simp le rep resen ta tion

of the c la ssic a l osc illa tor in d imen sion n, w ith a n ob v iou s c on ju g a tion c on d ition

imp osed . F or k = R, w e ha v e a rea l P la n ck ’s c on sta n t w hich w e ob v iou sly ma y

a ssu me eq u a l to = 1.

GEOMETRY OF TIME-SPACES. 49



Moreover, we now have, ([a+ , a− ]) = 1 , and we have a rep resentation of P h C

onto the alg eb ra F , g enerated b y {a+ , a− }. N otic e that in each fi nite rank , this

alg eb ra g enerate the whole E nd k (V ). T he c om m u tation relations is g iven b y a

c lassic al form u la,

n−2 m−2

am an = an am + mn an−1 am−1 + 1 /2!m(m − 1 )n(n − 1 ) a+ a−

− + + − + −



+ 1 /3 !m(m − 1 )(m − 2)n(n − 1 )(n − 2) an−3 am−3 + ...

+ −





and the L ie alg eb ra f, of d erivations of F are easily seen to b e g enerated b y the

d erivations {δp,q }p,q , d efi ned as,



p−1

δp,q (a+ ) = ap aq , δp,q (a− ) = −p/(q + 1 )a+ aq+1 .

+ − −





If we p u t, for m, n ≥ 0



χm,n := δm+1,n , χm := χm,0



then we fi nd the W itt-alg eb ra, with the c lassic al relations,



[χm , χn ] = (n − m)χm+n .



Moreover we fi nd ,



[χ0 , χm,n ] = (m − n)χm,n =: d e g (χm,n )χm,n .



C learly the L ie alg eb ra D e rk (F ) has an asc end ing fi ltration with resp ec t to the

d eg ree, d e g , d efi ned ab ove, and it is easy to see that the c orresp ond ing g rad ed L ie

alg eb ra g := g r(D e rk (F )) has the following p rod u c ts,



[χp,q , χr ,s ] = (r − p + (s + 1 )−1 (r + 1 )q − (q + 1 )−1 (p + 1 )s)χp+r ,q+s .



In p artic u lar the d eg ree z ero c om p onent of g is A b elian.

Example 3.11. F inally let C := R[x], and let C := C ⊗R C, and c onsid er som e

rep resentation on V = C of P h C = R . C learly ,



E xt1 (V, V ) = 0,

C





b u t, in g eneral,

E xt1 h C (V, V )

P



is infi nite d im ensional.

(i) C onsid er the free p artic le, i.e. the d y nam ic al sy stem , σ g iven b y ,



L = 1 /2 d x2 , σ : δ 2 x = 0,



and let V b e d efi ned b y letting d x ac t as the id entity . T hen we fi nd that,



∂

[δ] = 0, Q = .

∂x

50 OLAV ARNFINN LAUDAL



This means that [δ] d o es no t mo v e V in the mo d u li sp ac e o f V . The H amilto nian

Q d efi nes time, and

e xp (tQ)(f (x)) = f (x + t).

(ii) C o nsid er the same d y namic al sy stem, and let V b e d efi ned b y letting d x ac t

∂

as . Then w e fi nd that,

∂x

∂

[δ] = 0, Q = ( )2 .

∂x

A s ab o v e, [δ] d o es no t mo v e V in the mo d u li sp ac e. The H amilto nian Q d efi nes

time, and the time ev o lu tio n lo o k s lik e,



U (t, ψ) = e xp (tQ)(ψ).

ˆ

Intro d u c ing the F o u rier transfo rmed ψ, w e o b tain a time ev o lu tio n g iv en b y ,

ˆ ˆ

U (t, ψ) = e xp (tp 2 )(ψ).



(iii) C o nsid er ag ain the harmo nic o sc illato r, and let the rep resentatio n V := k[x−1 ]

∂

b e d efi ned b y letting x ac t as mu ltip lic atio n b y x−1 , and d x ac t as . Then w e

∂x

fi nd that,

∂

[δ] = 0, Q = (x ).

∂x

A s ab o v e, [δ] d o es no t mo v e V in the mo d u li sp ac e. The eig env ec to rs o f the H amil-

to nian Q are the mo no mials x−n , n ≥ 0, w ith eig env alu es −n, and the time ev o lu -

tio n lo o k s lik e,

U (t, x−n ) = e xp (−nt)x−n .

N o tic e that,

[x, d x] = x2 ,

as o p erato rs o n V . N o tic e also that V in this c ase is no t simp le. It is, ho w ev er, a

limit o f the fi nite rep resentatio ns, Vn := k[x−1 ]/(x−1 )n . The rep resentatio n V2 is

g iv en b y the ac tio ns,



0 0

x=

1 0

0 0

dx =

0 0



w here w e hav e cho sen the b asis {1, x−1 } in V2 . It is c learly no t simp le, b u t it sits

as a p o int at infi nity , t1 = t2 = 1 + t3 = t4 = 0, t5 = 1, fo r the (almo st) v ersal

family ,



0 1 + t3

x=

t5 t4

t1 t2

dx = .

1 + t3 0



F o r a tho ro u g h intro d u c tio n to the c lassic al Q u antu m Theo ry , u sing the theo ry

ab o v e, see the M aster thesis, and later p rep rints o f O lav G rav ir Imenes, [G I].

GEOMETRY OF TIME-SPACES. 51



Vertex-like algebras, and relations to the moduli space of algebraic curves. Given

a d y na m ic sy stem , A(σ) a nd a versa l fa m ily fo r n-d im ensio na l sim p le rep resenta -

tio ns. L et ξ b e th e fu nd a m enta l vec to rfi eld d efi ned o n U (n). R ec a ll T h eo rem (2 .1 3 )

a nd T h eo rem (3 .2 ). T h ere is a c o m m u ta tive d ia g ra m o f g enera liz ed sch em es,



U (n) / S im p n (A)





 

Yn := U (n)/ / S im p n (A)/ =: Xn .



T h e q u o tient “ sp a c es” Yn a nd Xn a re o rb it sp a c es, w h ere ea ch o rb it is a c u rve.

C o m p leting , w h en nessec a ry , U (n) a nd / o r S im p n (A), w e m a y a ssu m e th ese c u rves

c o m p lete. R estric ting to a n integrable part o f U (n), resp . o f S im p n (A), w e m a y

th en h o p e to fi nd na tu ra l m o rp h ism s,



Γn : Yn → M,



w h ere M is th e m o d u li sp a c e o f th e c o m p lete a lg eb ra ic c u rves. M o reo ver, T h eo rem

(3 .3 ) sh o u ld p ro d u c e a ra nk n b u nd le Vn o n Yn , n ≥ 1 , a nd o ne m ig h t a sk fo r

c o nd itio ns fo r th e ex istenc e o f u niversa l b u nd les Un o n M, su ch th a t Vn = Γ∗ Un . n

T h ese a re q u estio ns rela ted to vertex a lg eb ra s (b u nd les), see e.g . [F ]. T h ere is a

la rg e litera tu re o n th e su b jec t. S een fro m o u r p o int o f view , th e h id d en a g end a o f

th e vertex a lg eb ra fra m ew o rk , seem s to b e to c o nstru c t th e releva nt a lg eb ra A(σ)

o f o b serva b les fo r a g iven q u a ntu m (fi eld ) th eo retic situ a tio n.

In o u r la ng u a g e, let t0 ∈ S im p n (A(σ)) b e a sing u la rity fo r ξ. C o nsid er th e

P la nc ’s c o nsta nt, (t0 ), a nd th e c o rresp o nd ing o p era to rs, a+ , a− ∈ A(σ), to g eth er

i i

˜ ˜

w ith th e vacuum state ω(t0 ) ∈ V (t0 ) =: V (a ny fl a t sec tio n ω o f V a lo ng γ w ill

p ro d u c e a va c u u m sta te), su ch th a t th e a c tio n o f A(σ) ind u c es a n iso m o rp h ism ,



k[a+ , ..., a+ ]

1 r V,



a situ a tio n th a t w e h a ve seen rea liz ed in th e c a se o f th e h a rm o nic o sc illa to r in

d im ensio n 1 , b u t w h ich is ea sily seen to g enera liz e to a ny d im ensio n, th en th ere

p o p s u p a fa m ily o f g enera liz ed vertex a lg eb ra s. In fa c t, c o nsid er th e restric tio n o f

th e versa l fa m ily

ρ : A(σ) → E nd C(n) (V ),

˜ ˜



to th e integ ra l c u rve γ th ro u g h th e p o int t0 ∈ S im p n (A(σ)). It is sing u la r a t t0 , so

p a ra m etriz ed w ith tim e, τ , th e c o m p letio n w ill p ro d u c e a m a p ,

˜

Y : V = V (t0 ) k[a+ , ..., a+ ] ⊂ A(σ) → E nd k (V ) ⊗k k[[τ ]][τ −1 ],

1 r



see (3 .6 ), w h ich w ill b e a k ind o f generalized vertex algebra. In p a rtic u la r, th e

lo c a liz a tio n a x io m o f vertex a lg eb ra s im p ly th a t ρ(a+ ) a nd ρ(a+ ) c o m m u te, w h ich

˜ i ˜ j

h ere is o b vio u s. M o reo ver w e o b serve th a t th e exponentiating fo rm u la o f Y .-Z .

H u a ng , see [F ], p .1 8 , (1 6 ),



Y (A, t) = R(ρ)Y (R(ρ(t)−1 A, ρ(t))R(ρ)−1 ,

ˆ

fo r a ny ρ ∈ Au t(Oγ ,0 ) Au t(C[[t]]) fo llo w s fro m T h eo rem (3 .3 ) a b o ve. W e sh a ll,

h o p efu lly , retu rn to th is in a la ter p a p er.

52 OLAV ARNFINN LAUDAL



§4. Interactions. Given a dynamical system σ o f, o rder 2 . A particle, i.e. a

p o int t0 ∈ S im p n (A(σ)), (o r its co mp o nent) rep resented b y a p o int in U (n), a

˜

fi b er o f th e versal family, i.e. o f th e b u ndle V o n U (n) ⊂ S im p 1 (C(n)), w ill after

so me tim e τ h ave devello p ed into th e p article sitting at a p o int t1 o n th e integ ral

cu rve γ, defi ned b y th e vecto rfi eld ξ o f σ. N o w , th is may w ell b e a p o int o n th e

b o rder o f S im p n (A(σ)), i.e. in Γ n = S im p 1 (C(n)) − U (n), w h ere it d ecay s into an

indeco mp o sab le, o r into a semi-simp le, b u t no t simp le rep resentatio n, i.e. into tw o

o r mo re new p articles {Vi }.

W h at h ap p ens no w , is tak en care o f b y th e fo llo w ing scenario : If th e diff erent

p articles w e h ave p ro du ced are no t interacting , each o ne o f th e new p articles sh o u ld

b e co nsidered as an indep endent o b ject, w ith its o w n dynamical system defi ned b y

a L ag rang ian Li and a resu lting D irac derivatio n δi . H o w ever, if th e p articles w e

h ave p ro du ced are interacting , w e h ave a diff erent situ atio n.

N o tice fi rst th at fo r n = 1 , w e h ave a cano nical mo rp h ism o f sch emes,



S im p 1 (A(σ)) −→ S im p 1 (A)



and a cano nical vecto r-fi eld ξ in S im p 1 (A(σ)), th e ph ase space. Given any p o int

o f S im p 1 (A), th e co n fi g u ratio n space, and any tang ent-vecto r at th is p o int, th ere is

an integ ral cu rve o f ξ in S im p 1 (A(σ)), th ro u g h th e co rresp o nding p o int, p ro jecting

o nto th e fu n d am en tal cu rv e in th e co nfi g u ratio n sp ace.

F o r n ≥ 2 th e sp aces S im p n (A(σ)) and S im p n (A) are, h o w ever, to tally diff erent

and w ith o u t any easy relatio ns to each o th er.

L et no w vi ∈ S im p ni (A(σ)), i = 1 , 2 b e tw o p o ints o f S im p (A(σ)), mayb e in

diff erent co mp o nents, and/ o r rank s. C o nsider th eir co mp o nents, i.e. th e u niversal

families in w h ich th ey are rep resented,



ρi : A(σi ) −→

˜ ˜

E nd C(ni ) (Vi )



T h e D irac derivatio n, δi defi nes derivatio ns,



[δi ] : A(σ) −→ ˜

E nd C(ni ) (Vi )



and th erefo re also th e fu ndamental vecto r-fi elds,



ξi ∈ D e r k (C(ni )).



4.1 D efi nition. L et B be an y fi n itely g en erated k-alg eb ra. W e sh all say th at th e

˜

co m po n en ts, C1 ⊆ S im p n1 (B), C2 ⊆ S im p n2 (B), o r th e co rrespo n d in g particles Vi ,

i= 1 ,2 , are n o n -in teractin g if



E x t1 (V1 , V2 ) = 0 , ∀v1 ∈ C1 , ∀v2 ∈ C2 .

B





O th erw ise th ey in teract.

S u p p o se no w th at th e p o ints v1 and v2 , sit in S im p n1 (A(σ)) and S im p n2 (A(σ)),

resp ectively. ” P h ysically” , w e sh all co nsider th is as an ” o b servatio n” o f tw o p arti-

˜ ˜

cles, V1 and V2 in th e ” state” V1 and V2 , at so me instant. If th e tw o p articles are

no n-interacting , th e resu lting entity, co nsidered as th e th e su m V := V1 ⊕ V2 , o f

GEOMETRY OF TIME-SPACES. 53



dimension n := n1 + n2 , a s modu le ov er A(σ), w ill sta y , ” a s time p a sses” , a su m of

tw o simp les.

If V1 a nd V2 intera c t, th is ma y ch a ng e. T o ex p la in w h a t ma y h a p p en, w e h a v e

to ta k e into c onsidera tion th e non-c ommu ta tiv ity of th e g eometry of P h A . In

p a rtic u la r, w e h a v e to c onsider th e non-c ommu ta tiv e deforma tion th eory , see [L a

2,3 ,4 ]. C onsider th e deforma tion fu nc tor,

D e f {V1 ,V2 } : a2 −→ S e ts ,

or, if w e w a nt to dea l w ith more p oints, sa y a fi nite fa mily Vi , i = 1 , 2, ..., r th e

deforma tion fu nc tor,

D e f {Vi } : a2 −→ S e ts ,

a nd its forma l modu li,  

H1,1 ... H1,r

H :=  . ... . .

Hr,1 ... Hr,r

tog eth er w ith th e v ersa l fa mily , i.e. th e essentia lly u niq u e h omomorp h ism of k-

a lg eb ra s,

 

H1,1 ⊗ E nd k (V1 ) ... H1,r ⊗ Ho m k (V1 , Vr )

˜

ρ : A(σ) −→  . ... . .

Hr,1 ⊗ Ho m k (Vr , V1 ) ... Hr,r ⊗ E nd k (Vr )

W e need a w a y of sp ec ify ing w h ich intera c tions w e w a nt to c onsider. T h is is th e

p u rp ose of th e follow ing , tenta tiv e, defi nition,

4 .2 D e fi n itio n . Let B be a n y fi n itely g en era ted k-a lg eb ra . W e sh a ll sa y th a t w e

h a v e sp ec ifi ed a n in tera c tio n m o d e in S im p (B), if w e h a v e g iv en a q u iv er Γ, w ith

v ertices {vi }, i = 1 , ..., r, a n d , fo r a n y c h o ice o f a fi n ite fa m ily o f sim p le B-m o d u les

{Vi }, i= 1 ,...,r, a h o m o m o rp h ism ,

φ : H({Vi }) −→ kΓ.



A n intera c tion mode is a k ind of h ig h er o rd er p rep a ra tio n , see (1 .2). In fa c t, a n

intera c tion mode c onsists of a ru le, telling u s, for ea ch fa mily of r p oints {vi }, vi ∈

S im p ni (B) a nd ea ch p resc rib ed seq u enc e of B-modu les {Vi }i= 1,...,n rep resenting

some of th ese th ese p oints, h ow to p rep a re th eir intera c tions. T h e morp h ism φ fi x es

a ll relev a nt h ig h er o rd er momenta , i.e. φ ev a lu a tes a ll th e ta ng ents b etw een th e

modu les, a nd c rea tes a new B-modu le.

In fa c t, a n intera c tion mode indu c es a h omomorp h ism,

 

kΓ(1,1) ⊗ E nd k (V1 ) ... kΓ(1,r) ⊗ Ho m k (V1 , Vr )

˜

ρα : B −→  ... .

kΓ(r,1) k ⊗ Ho m k (Vr , V1 ) ... kΓ(r,r) ⊗ E nd k (Vr )

A nd th e la st ma trix a lg eb ra is na tu ra lly emb edded in,

E nd k (V ) Mk (n),

w h ere V := ⊕i= 1,...,r Vi . T h u s, w e h a v e c onstru c ted a new B-modu le, w h ich ma y b e

dec omp osa b le, indec omp osa b le or simp le, dep ending on th e intera c tion mode w e

ch oose, a nd, of c ou rse, dep ending u p on th e ta ng ent stru c tu re of th e modu li sp a c e

S im p (B).

54 OLAV ARNFINN LAUDAL



Example 4.3. Let B b e th e free k-a lg eb ra o n tw o n o n -c o m m u tin g sy m b o ls, B =

k , a n d see E x a m p le (2.14 ). Let p1 a n d p2 b e tw o d iff eren t p o in ts in

th e (x1 , x2 )-p la n e, a n d let th e c o rresp o n d in g sim p le B-m o d u les b e V1 , V2 . T h en ,

E xt1 (V1 , V2 ) = k. Let Γ b e th e q u iv er,

B





V1 • / • V2 ,



th en a n in tera c tio n m o d e is g iv en b y th e fo llo w in g elem en ts: F irst th e fo rm a l m o d u li

o f {V1 , V2 },

k

H := ,

k

th en th e k-a lg eb ra ,

k k

kΓ := ,

0 k

a n d fi n a lly a h o m o m o rp h ism ,

φ:H− → kΓ

sp ec ify in g th e v a lu e o f φ(t1,2 ) ∈ E xt1 (V1 , V2 ). S in c e Ho m k (Vi , Vj ) = k, w e o b ta in

B

V = k 2 , a n d w e m a y ch o o se a rep resen ta tio n o f φ(t1,2 ) a s a d eriv a tio n , ψ1,2 ∈

D e r k (B, Ho m k (V1 , V2 )), su ch th a t th e B-m o d u le V = k 2 is d efi n ed b y th e a c tio n s

o f x1 , x2 , g iv en b y ,



α1 1 β1 0

X1 := , X2 := ,

0 α2 0 β2



w h ere p1 = (α1 , β1 ) a n d p2 = (α2 , β2 ). V is th erefo re a n in d ec o m p o sa b le B-m o d u le,

b u t n o t sim p le.

If w e h a d ch o sen th e fo llo w in g q u iv er,



2,1

V1 • o / • V2 , i,j j,i = 0, i, j = 1, 2,

1,2









th en th e resu ltin g B-m o d u le V = k 2 w o u ld h a v e b een sim p le, rep resen ted b y ,



α1 1 β1 0

X1 := , X2 := .

0 α2 1 β2



In g en era l, if B = A(σ), a n d if (σ) is d efi n ed b y a La g ra n g ia n , w ith D ira c

d eriv a tio n δ, a n y in tera c tio n m o d e p ro d u c in g a sim p le m o d u le V , th u s a p o in t

v ∈ S im p(A(σ)), rep resen ts a creation o f a n ew p a rtic le fro m th e in fo rm a tio n

c o n ta in ed in th e in tera c tin g c o n stitu en c ies. M o reo v er, a n y fa m ily o f sta te-v ec to rs

ψ ∈ Vi , p ro d u c es a c o rresp o n d in g sta te -v ec to r ψ := i= 1,... ψi ∈ V , a n d T h eo rem

(3 .3 ) th en tells u s h o w th e ev o lu tio n o p era to r a c ts o n th is n ew sta te-v ec to r. If th e

c rea ted n ew p a rtic le V is n o t sim p le, th e D ira c d eriv a tio n δ ∈ D e r k (A(σ)), w ill

in d u c e a ta n g en t v ec to r [δ](V ) ∈ E xt1 h A (V, V ) w h ich m a y o r m a y n o t b e m o d u la r,

P

o r p ro -rep resen ta b le, w h ich m ea n s th a t th e p articles Vi , w h en in teg ra ted in th is

d irec tio n , m a y o r m a y n o t c o n tin u e to ex ist a s d istin c t p a rtic les, w ith a n o n -triv ia l

en d o m o rp h ism rin g , o r, w ith a Lie a lg eb ra o f a u to m o rp h ism s, eq u a l to k 2 . If th ey

GEOMETRY OF TIME-SPACES. 55



do, this situation is analogous to the case which in physics is referred to as the

” super-selection rule” .

O r, if [δ] ∈ E x t1 (V, V ) does not sit (or stay) in the m odular stratum , the

A(σ)

particle V looses autom orphism es, and m ay b ecom e indecom posib le, or sim ple,

instantaneously.

W e m ay thus create new particles, and we hav e in E x am ple (3 .6 ) discussed the

notion of lifetim e for a giv en particle. In particular we found that the harm onic

oscillator had ev er-lasting particles of rank 2 . If, howev er the L agrangian of the

system had b een diff erent, we m ay produce particles of fi nite lifetim e.

Example 4.4. L et, as in (3 .6 ) A(σ) = P h A = k , and consider the

curv e of two-dim ensional sim ple P h A-m odules,



0 1+ t

X1 =

0 t

t 0

X2 = .

1+ t 0



C om puting the F orm anek center f , see (3 .6 ), we fi nd,



f (t) = t2 (1 + t)2 − (1 + t)4 .



T he corresponding particle, b orn at t = 0, decays after τ = −1/2 , and thus has a

fi nite lifetim e.

G iv en a k-algeb ra A we hav e therefore forged a fram ework for creating standard

m o de ls, for all fi nite fam ilies of particles of dynam ical system s of order 2 . W e ” just”

tak e all interesting particles in (com ponents of) S im p n (A(σ)), n ≥ 1, call them

{Vi }i= 1,2,.. , we choose corresponding relativ e L agrangians, or dynam ical system s of

order 2 , determ ining their D irac deriv ations δi ∈ D e r k (A(σ)), and then we com pute

all interactions, all decays, all possib le creations, etc. and then, m ayb e, cook up

ne w co m m o n L agrangians, for the particles the diff erent interaction m odes create.

W e shall (hopefully) return to concrete ex am ples of such ” standard m odels” in a

later paper.



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Box.1053, Blindern 0316 Oslo, Norway

E-mail address: arnfinnl@math.uio.no