# On TimelikeExcitations in the Relativistic Harmonic Oscillator

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```					                   On Timelike Excitations
in the
Relativistic Harmonic Oscillator

Martin Land
IARD 2008

Jerusalem

June 2008

Martin Land — IARD 2008         Relativistic Harmonic Oscillator   June 2008   1 / 33
Relativistic Harmonic Oscillator
Hamiltonian approach — formally similar to nonrelativistic oscillator
p2
+ V (x) ψ (x) = κ ψ (x)                                        (1)
2m

1          1
p2 = η µν pµ pν         V (x) =     mω 2 x2 = mω 2 x2 − t2                η µν = diag (−1, 1, 1, 1)
2          2
Three related but diﬀerent approaches
Feynman, Kislinger, Ravndal, Current Matrix Elements from a Relativistic Quark
Model (1971)
Y. S. Kim and M. Noz, Covariant Harmonic Oscillators and the Quark Model (1973)
Arshansky and Horwitz, The Quantum Relativistic Two-Body Bound State (1989)
Issues requiring care and attention
Lorentz covariance of states
Indeﬁnite spectrum for timelike separation (x2 < 0)
Normalizability of states

Martin Land — IARD 2008              Relativistic Harmonic Oscillator                  June 2008    2 / 33
The approach of Feynman et. al.
Apply methods that work well in 3D nonrelativistic case

Dimensionless coordinates
√                                       1
qµ =       mω x µ                       π µ = √mω pµ                                         (2)

Dirac’s factorization of 1D Hamiltonian into ﬁrst order operators
1              1                 1                       1
K=ω N+      2   = ω aa +   2          a=     √
2
(q + iπ )    a=   √
2
(q − iπ )

Separation of variables in Cartesian coordinates
ai , a j = δij −→ [ aµ , aν ] = η µν               |n −→ n0             n1       n2        n3               (3)

K −→ ωη µν aµ aν + 1 η µν = ω − N 0 + N 1 + N 2 + N 3 + 2
2                                                                                        (4)

Scalar ground state function
i j                  −                  η µν qµ qν /2
ψ0 (q) = e−(δij q q )/2 −→ ψ0 (q) = e                                                             (5)
Ground state annihilated by lowering operators
a µ ψ0 ( q ) = 0                                                             (6)

Martin Land — IARD 2008                Relativistic Harmonic Oscillator                                    June 2008   3 / 33
Closer look at covariant number representation
Operators
1                                        1
aµ = √ (qµ + iπ µ )                      aµ = √ (qµ − iπ µ )                                       (7)
2                                            2
Commutation relations
[ aµ , aν ] = η µν              [ N µ , aν ] = η µν aµ                   [ N µ , aν ] = −η µν aµ                (8)

Raising and lowering operators
µ                                                √         µ
aµ |n =         nµ + η µµ eiφ+ |n + η µν eν                          aµ |n =              nµ eiφ− |n − η µν eν           (9)

e ν = δ0 , δ1 , δ2 , δ3
ν ν ν ν

Choice of phase determines structure of ground state
0          0                 k             k
Feynman:            eiφ+ = eiφ− = i, eiφ+ = eiφ− = 1

a0 | n =          1 − n0 | n − e0                a0 | n =             −n0 |n + e0 −→ n0 ≤ 0                     (10)
µ             µ
Kim:        eiφ+ = eiφ− = 1
√
a0 | n =             n0 − 1 | n − e0              a0 | n =               n0 |n + e0 −→ n0 ≥ 1                  (11)

Martin Land — IARD 2008                        Relativistic Harmonic Oscillator                                 June 2008    4 / 33
Norm and spectrum
Feynman phase
Indeﬁnite norm states
√
a0 |0 0 0 0 =       1 |−1 0 0 0                 a0 |0 0 0 0 = 0               (12)

nµ | nν = 0 0 0 0| aµ aν |0 0 0 0 = 0 0 0 0| aν aµ + η µν |0 0 0 0 = η µν                         (13)

Positive spectrum
− n0 n1 n2 n3 | K | − n0 n1 n2 n3 = ω ( n0 + n1 + n2 + n3 + 2)                          (14)
Kim phase
Positive norm states
a0 |1 0 0 0 = 0             a0 |1 0 0 0 = |2 0 0 0                         (15)

nk | nl = 1 0 0| ak al |1 0 0 = η kl                                  (16)

n0 | n0 = 1 0 0 0| a0 a0 |1 0 0 0 = 1 0 0 0| a0 a0 − η 00 |1 0 0 0 = 1                         (17)

Indeﬁnite spectrum
n0 n1 n2 n3 |K |n0 n1 n2 n3 = ω (−n0 + n1 + n2 + n3 + 2)                              (18)

Martin Land — IARD 2008                Relativistic Harmonic Oscillator                     June 2008    5 / 33
Problems of interpretation in the Feynman approach

Ground state not normalizable
2          − η µν qµ qν                2
− t2 )
ψ0 ( q )        =e                       = e−(q                                     (19)

Timelike excitations can lead to indeﬁnite norm
−1,      n0 > 0
| ψ |2 = − n0 n1 n2 n3 | − n0 n1 n2 n3 =                                                              (20)
1,       n0 = 0
Ad hoc suppression of timelike excited states
Only calculate with states satisfying
( p · a) |ψ = 0                                              (21)

Still too many degrees of freedom associated with ground state
1              1
K ψ0 (q) = η µν η µν ω ψ0 (q) = 4 × ω ψ0 (q)                                           (22)
2              2
Ad hoc suppression of timelike states −→ missing states in matrix elements

Martin Land — IARD 2008                     Relativistic Harmonic Oscillator                         June 2008    6 / 33
The approach of Kim and Noz

Normalizable solution for ground state
2  2         2      2
ψ0 (q) = e−(q +t )/2 = e−q /2 e−t /2                        (23)
µ             µ
Equivalent to choice of phase:             eiφ+ = eiφ− = 1
√
a0 | n =        n0 − 1 | n − e0    a0 |n = n0 |n + e0 −→ n0 ≥ 1               (24)

2
Trial solution φ0 (q) = H (q) e−q             /2

1                    2  2      1
ak ϕ0 (q) = √ x k + ∂k H (q) e−(q −t )/2 = √ ∂k H (q) = 0                (25)
2                              2
1                     2  2      1
a0 ϕ0 (q) = √ (t + ∂t ) H (q) e−(q −t )/2 = √ (2t + ∂t ) H (q) = 0              (26)
2                               2
Kim-Noz ground state is solution to ﬁrst order annihilation equations
2                2  2
H (q) = e−t −→ ϕ0 (q) = e−(q +t )/2                        (27)

Martin Land — IARD 2008              Relativistic Harmonic Oscillator       June 2008    7 / 33
Covariant Bound States
Many-body formalism — Horwitz and Piron (1973)

∂                     ( p1 ) 2 ( p2 ) 2
i      ψ ( x1 , x2 , τ ) =         +         + V ( x1 , x2 ) ψ ( x1 , x2 , τ )                             (28)
∂τ                      2M1      2M2
Central force problem — Horwitz and Arshansky (1989)

( p1 ) 2 ( p2 ) 2                   Pµ Pµ   pµ pµ
+         + V ( x1 , x2 ) =       +       + V x2                                             (29)
2M1      2M2                        2M      2m
µ        µ
P µ = p1 + p2                 M = M1 + M2

µ             µ
pµ = M2 p1 − M1 p2 /M                          m = M1 M2 /M

x = x1 − x2        x 2 = ( x1 − x2 )2 − ( t1 − t2 )2           x2 → x2 = r2 in Galilean limit

Eﬀective one-body oscillator eigenvalue problem
p2  1
ψ ( x, τ ) = ψ ( x ) e−iκτ                        + mω 2 x2 ψ ( x ) = κ ψ ( x )                            (30)
2m  2

Martin Land — IARD 2008                      Relativistic Harmonic Oscillator                              June 2008    8 / 33
A priori spacelike support

Require spacelike separation
p2  1
K ψ (q) =            + mω 2 x2 ψ ( x ) = κ ψ ( x )                 x 2 = x2 − t2 > 0           (31)
2m  2
Virial theorem ⇒ K > 0

p2          1 µ              1
=     x ∂µ V ( x ) =   mω 2 x2 = V ( x ) > 0                                (32)
2M           2                2

No obvious way to realize nonholonomic constraint in Cartesian coordinates
Hyperspherical parameterization

ˆ
x = ρx       ρ=       x2 − t2              x 2 = x2 − t2 = 1
ˆ     ˆ    ˆ                          (33)
3
O(3,1) bound state solutions with K > 0 and K ψ0 = 2 ω ψ0
How do these states appear in the number representation?
How do creation/annihilation operators act on these states?
How are the timelike occupation number modes suppressed?

Martin Land — IARD 2008              Relativistic Harmonic Oscillator                      June 2008    9 / 33
Program
Study O(3) nonrelativistic and O(2,1) covariant cases
Symmetries of same dimension permits easier comparison
O(3,1) case involves induced representation of O(3,1) over O(2,1)
Solve eigenvalue equation K ψ = κ ψ in (hyper)spherical coordinates
Maximal commuting operator set — K, Casimir operator, one generator
Separation of variables exploiting symmetry operators
Characterize states by eigenvalues of commuting operators
Extract eigenvalue information from number states
Express Hamiltonian and symmetry operators in terms of operators aµ and aµ
Study how number states participate in (hyper)spherical eigenstates

Results
O(3) oscillator — obtain energy and angular momentum eigenvalues from
number representation
O(2,1) oscillator — hyperspherical solution not related to number states by
any unitary transformation
Martin Land — IARD 2008     Relativistic Harmonic Oscillator        June 2008   10 / 33
Harmonic Oscillator in 3 Dimensions
Solution in polar coordinates

Eigenvalue equation
ω
Kψ=        π 2 + q2 ψ = Eψ                                                (34)
2
Coordinates
q = r (sin θ cos φ, sin θ sin φ, cos θ ) −→ q2 = r2                                 (35)

O(3) generators and Casimir operator
1 ijk k l
Li =      ε   q π − ql π k = εijk q j π k                     L2 , K = L3 , K = 0               (36)
2
cos θ     1                  2
L3 = −i∂φ        L2 = − ∂2 −                ∂ +               L3                       (37)
θ            sin θ θ sin2 θ
Momentum
2     1
π2 = −      2
= − ∂2 − ∂r + 2 L2
r                                                       (38)
r    r
Solution
1 − r2 l l + 1 2                             3
ψnlm =          e 2 r Ln 2 r Ylm (θ, φ) −→ E = ω 2n + l +                                        (39)
Nnl                                           2

Martin Land — IARD 2008              Relativistic Harmonic Oscillator                         June 2008    11 / 33
Harmonic Oscillator in 3 Dimensions
Number representation

Operators
1                                  1
ak = √ qk + iπ k                   ak = √ qk − iπ k                                  (40)
2                                  2
ak , a j = δkj            k, j = 1, 2, 3                                   (41)

Hamiltonian
1                               3                                   3
K     =     ω ak ak + δk       = ω a·a+                      = ω N1 + N2 + N3 +
2 k                             2                                   2
3
=     ω N+                                                                                      (42)
2
States
N k | n = a k a k | n = n k | n = n k n1 n2 n3                         K |n = ω (n + 3/2) |n              (43)

ak |n =       nk + δkk n + δkj e j                 ak |n =        nk n − δkj e j               (44)

e j = δ1 , δ2 , δ3
j j j                                                         (45)

Martin Land — IARD 2008              Relativistic Harmonic Oscillator                          June 2008    12 / 33
Harmonic Oscillator in 3 Dimensions
Spherical operators in number representation

Spherical operators
a± = a1 ± ia2               a± = a1 ± ia2                               (46)
[ a+ , a+ ] = [ a− , a− ] = 0            [ a+ , a− ] = [ a− , a+ ] = 2               (47)

Number operator
1
N=   ( a + a − + a − a + ) + a3 a3                                     (48)
2
3    1                             3
K = N + = ( a + a − + a − a + ) + a3 a3 +                                    (49)
2    2                             2

Angular momenta
L2 = N2 + N − ( a · a ) ( a · a )                                     (50)
1
L3 =      ( a+ a− − a− a+ )                                        (51)
2

Martin Land — IARD 2008                Relativistic Harmonic Oscillator                    June 2008    13 / 33
Harmonic Oscillator in 3 Dimensions
Coherent subspaces

Commutation relations
N 3 , a ± = N 3 , a ± = N 3 , L3 = a3 , L3 = a3 , L3 = 0                                 (52)

L2 , N i = 0             L3 , N 1 = 0                 L3 , N 2 = 0                (53)

L3 is block diagonal in n and n3
3 3
n1 n2 n3 L3 n1 n2 n3                  = c n1 , n2 , n3 δnn δn      n
(54)

L3 is Hermitian and preserves n = n1 + n2 + n3 and n3

L3 n1 n2 n3         =     i      n1 ( n2 + 1) n1 − 1 n2 + 1 n3

−i         n1 + 1 n2 n1 + 1 n2 − 1 n3                   (55)

Martin Land — IARD 2008                  Relativistic Harmonic Oscillator                   June 2008    14 / 33
Harmonic Oscillator in 3 Dimensions
Multiplicity of states

Label states n1 n2 n3 by n and n3

k      n − n3 − k        n3 ,     k = 0, 1, ..., n − n3 ,               n3 = 0, 1, ..., n           (56)

Multiplicity of states for given n
n
( n + 1) ( n + 2)
∑      n − n3 + 1 =
2
(57)
n3 =0
Deﬁne states
1                                  γ
ζ αβγ =        ( a + ) α ( a − ) β a3             |0                               (58)
Nαβγ
where N αβγ is a normalization factor, and
α+β+γ = n                                                           (59)
On these states
1 1
L3 ζ αβγ =            ( a+ a− − a− a+ ) ( a+ )α ( a− ) β ( a3 )γ |0 = (α − β) ζ αβγ                      (60)
2 Nαβγ

Martin Land — IARD 2008               Relativistic Harmonic Oscillator                            June 2008    15 / 33
Harmonic Oscillator in 3 Dimensions
Total angular momentum

The total angular momentum
1                      1
L2 ζ αβγ =        N (N + 1) ζ αβγ −      ( a · a) ( a · a) ζ αβγ                  (61)
Nαβγ                   Nαβγ
On the states ζ αβγ

L2 ζ αβγ     =    [(α + β + γ) (α + β + γ + 1) − 4αβ − γ (γ − 1)] ζ αβγ
N(α+1)( β+1)(γ−2)
−                           γ (γ − 1) ζ (α+1)( β+1)(γ−2)
Nαβγ
N(α−1)( β−1)(γ+2)
−                           4αβζ (α−1)( β−1)(γ+2)                      (62)
Nαβγ

Martin Land — IARD 2008               Relativistic Harmonic Oscillator               June 2008    16 / 33
Harmonic Oscillator in 3 Dimensions
Maximum angular momentum for number state

States ζ αβγ have n = α + β + γ and m = α − β but are mixtures of l -states

The cases (α, β, γ) = (n, 0, 0) and (α, β, γ) = (0, n, 0) have eigenvalues
L2 ζ n00 = n (n + 1) ζ n00               L3 ζ n00 = nζ n00               (63)

L2 ζ 0n0 = n (n + 1) ζ 0n0              L3 ζ 0n0 = −nζ n00                (64)

The allowed eigenvalues of L3
m = α − β = −l, −l + 1, ..., l − 1, l                               (65)

are consistent with
α, β = 0, 1, ..., lmax = n                                    (66)

Martin Land — IARD 2008                  Relativistic Harmonic Oscillator               June 2008    17 / 33
Harmonic Oscillator in 3 Dimensions
Angular momentum content of states

L2 mixes (α, β, γ)-states with (α ± 1, β ± 1, γ                      2)-states

The mixed states have the same eigenvalues of L3
m = α − β = ( α ± 1) − ( β ± 1)                                          (67)

The angular momentum content of the states ζ αβγ is

n                lmax , lmax − 2, ..., 0   lmax even
l = n, n − 2, n − 4, ..., n − 2 · int             =                                                     (68)
2                lmax , lmax − 2, ..., 1    lmax odd

Since the multiplicity of l -states is 2l + 1 the total multiplicity of n-states is
int(n/2)
( n + 1) ( n + 2)
∑        [2 (2k) + 1] =
2
(69)
k =0

as required

Martin Land — IARD 2008                Relativistic Harmonic Oscillator                   June 2008    18 / 33
Harmonic Oscillator in 3 Dimensions
Sum of eigenvalues

For given values of n = α + β + γ and m = α − β, the possible (α, β, γ)-states are
n−m
α = m + k,            β = k,             γ = n − m − 2k,                         k = 0, 1, 2, ..., int                     (70)
2
The on-diagonal elements of L2 for given n and m are
L2 on−diagonal = (2n − m) (m + 1) + 2k (2n − 4m − 1) − 8k2                                                   (71)

Since tr L2 is invariant under diagonalization, the sum of eigenvalues is
int( n−m )
2
tr L2      =        ∑        L2 on−diagonal
k =0
    1 3
    6n    + 3 n2 + 5 n − 1 m3 + 1 m2 + 6 m
1
4      6     6      4                                 ,   n − m even
=                                                                                                    (72)
1 3       3 2       5        1 3       1 2       1        1
6n    +   4n    +   6n   −   6m    −   4m    +   6m   +       ,   n − m odd

4

Martin Land — IARD 2008                      Relativistic Harmonic Oscillator                                 June 2008    19 / 33
Harmonic Oscillator in 3 Dimensions
Classiﬁcation of states

For given values of n = α + β + γ and m = α − β, the possible l -states are
n−m
l = n − 2k               k = 0, 1, ..., int                                                (73)
2
Therefore, for given n and m the sum of eigenvalues is
int( n−m )
2
tr L2      =        ∑        (n − 2k) (n − 2k + 1)
k =0
    1 3
    6n    + 3 n2 + 5 n − 1 m3 + 1 m2 + 6 m
1
4      6     6      4                            ,   n − m even
=                                                                                               (74)
1 3
6n
1
+ 3 n2 + 5 n − 1 m3 − 1 m2 + 6 m +         1
,   n − m odd

4      6     6      4                    4

as required
Regarding k as a principal quantum number n a the total mode number becomes
3
n = 2 n a + l −→ E = ω 2n a + l +                                                          (75)
2

Martin Land — IARD 2008                     Relativistic Harmonic Oscillator                             June 2008    20 / 33
Harmonic Oscillator in 3 Dimensions
Unitary transformation for total mode number n = 2

1 − r2 l l + 2 2
1
ψnlm =
Nnl
e 2 r Ln    r Ylm (θ, φ) =   ∑ Cn1 n2 n3
nlm
Hn1 (r sin θ cos φ) Hn2 (r sin θ sin φ) Hn3 (r cos θ )

l      m     Unitary combinations of number states n1 n2 n3

1
0      0               √      |0 0 2 + |0 2 0 + |2 0 0
3

−2            − 1 |0 2 0 + 1 |2 0 0 +
2          2
1
√ i |1   10
2

1               1
−1                      √ i |0    1 1 + √ |1 0 1
2                           2

2                   1                      1
2      0          −    3   |0 0 2 +        √     |0 2 0 +         √     |2 0 0
6                      6

1                       1
1                     − √ i |0 1 1 +            √       |1 0 1
2                      2

−2            − 1 |0 2 0 + 1 |2 0 0 −
2          2
1
√ i |1   10
2

Martin Land — IARD 2008              Relativistic Harmonic Oscillator                              June 2008     21 / 33
Harmonic Oscillator in 2+1 Dimensions
Hyperspherical coordinates

Eigenvalue equation
ω
Kψ=      π 2 + q2 ψ = κ ψ                                                 (76)
2
Coordinates
q = (t, x, y) = ρ (sinh β, cosh β cos φ, cosh β sin φ) −→ q2 = ρ2                              (77)

O(2,1) generators and Casimir operator

L = q1 π 2 − q2 π 1      A1 = q0 π 1 − q1 π 0                A2 = q0 π 2 − q2 π 0               (78)

sinh β        L2
L = −i∂φ         Λ = L2 − A2 = ∂2 +                       ∂β +                             (79)
cosh2 β
β                  cosh β

L, A1 = iA2          L, A2 = −iA1                      A1 , A2 = −iL                      (80)

Momentum
2      Λ
π2 = −        2
= − ∂2 − ∂ ρ + 2
ρ                                                      (81)
ρ     ρ

Martin Land — IARD 2008             Relativistic Harmonic Oscillator                          June 2008    22 / 33
Harmonic Oscillator in 2+1 Dimensions
Separation of variables

Two separation parameters
ρ2             2                               1
− ∂2 − ∂ ρ + ρ2 − ε R ( ρ ) = −              Λb ( β) Φ (φ) = −Λ1                             (82)
R (ρ)        ρ
ρ                         b ( β) Φ (φ)

cosh2 β                sinh β                      1
∂2 +          ∂ − Λ1 b ( β ) = −       L2 Φ ( φ ) = − Λ2
2                       (83)
b ( β)          β     cosh β β                  Φ (φ)

φ-equation

L2 Φ ( φ ) = − ∂2 Φ ( φ ) = Λ2 Φ ( φ )
φ            2                        −→     Φ (φ) = eiΛ2 φ               (84)

β-equation
sinh β              Λ2
∂2 +          ∂ β − Λ1 +     2
b ( β) = 0                    (85)
cosh2 β
β     cosh β

Martin Land — IARD 2008                 Relativistic Harmonic Oscillator                     June 2008    23 / 33
Harmonic Oscillator in 2+1 Dimensions
Hyperspherical function

Change of variables
1
4
z = tanh β         b ( z ) = 1 − z2               P (z)                        (86)

Set constants of integration
1                      1          1
Λ1 = µ2 −               Λ2 = ν +            = µ + k + , k = 0, 1, 2, ...                        (87)
4                      2          2
β-equation becomes associated Legendre

µ2
1 − z2 ∂2 − 2z∂z + ν (ν + 1) −
µ
z                                                 Pν (z) = 0                     (88)
1 − z2
Eigenfunction for Λ and L
1
−µ                                         −µ                  1
Pµ+k (z) ei(µ+k+ 2 )φ
4
χµ+k ( β, φ) = Cµk 1 − z2                                                                (89)

−µ                      −µ                              −µ                      −µ
Λ χµ+k ( β, φ) = Λ1 χµ+k ( β, φ)                        L χµ+k ( β, φ) = Λ2 χµ+k ( β, φ)               (90)

Martin Land — IARD 2008                   Relativistic Harmonic Oscillator                        June 2008    24 / 33
Harmonic Oscillator in 2+1 Dimensions
Action of O(2,1) boost operators

Boost operators

A± = A1 ± iA2 = e±iφ −i 1 − z2 ∂z ± z∂φ                                                      (91)

Action of lowering operator on lowest state
1
−µ                                                                                    −µ
A− χµ ( β, φ) = e−iφ −i 1 − z2 ∂z − z∂φ
4
1 − z2          Pµ (z) Φµ (φ) = 0         (92)

Action of raising operator
−µ                                          −µ
A+ χµ ( β, φ) = Cµk i (2µ + 1) χµ+1 ( β, φ)                                             (93)

(2µ+k)!
Generally with Cµk =                µ      k!

−µ                                                         −µ
A+ χµ+k ( β, φ) = i             (k + 1) (2µ + k + 1)χµ+k+1 ( β, φ)                                (94)

Martin Land — IARD 2008                      Relativistic Harmonic Oscillator                          June 2008    25 / 33
Harmonic Oscillator in 2+1 Dimensions

1
2                µ2 −
∂2 +       ∂ ρ − ρ2 + ε −                 4
R (ρ) = 0                 (95)
ρ
ρ                  ρ2
Change of variables
x  2n−1
x = ρ2         R (ρ) = e− 2 x 4 L ( x )                               (96)

ρ -equation becomes Laguerre

∂2                 ∂      µ
x       + [ µ − x + 1]    + n Ln ( x ) = 0                                (97)
∂x2                ∂x

1 1
n=        ε−µ−1                 −→          κ = ω (2n + µ + 1)                  (98)
2 2
Solution
2
1 µ                          1      −µ                  1
(tanh β) ei(µ+k+ 2 )φ
ρ
ψ (ρ, β, φ) = Nnµk e− 2 ρµ− 2 Ln ρ2                           P                                   (99)
cosh β µ+k

Martin Land — IARD 2008                Relativistic Harmonic Oscillator                 June 2008    26 / 33
Harmonic Oscillator in 2+1 Dimensions
Ground state function

−µ                          µ
Taking n = µ = k = 0 ⇒ Pµ+k −→ P0 = 1, Ln −→ L0 = 1

1        ρ2    1
ψ0 (ρ, β, φ) = N0                    e − 2 ei 2 φ                          (100)
ρ cosh β
Ground state energy (mass)
1
κ = ω (2n + µ + 1) −→ ω = 2 ×                  2ω                           (101)

In Cartesian coordinates q = (t, x, y)
1                                               y
ρ cosh β = x2 + y2   4         ρ2 = x 2 + y2 − t2       φ = arctan
x

1                 2  2  2      i         y
ψ0 (t, x, y) = N0                           e−( x +y −t )/2 e 2 arctan( x )               (102)
( x2   + y2 )1/4
Satisﬁes Cartesian eigenvalue equation
1
2    −∂µ ∂µ + qµ qµ        ψ0 (t, x, y) = ψ0 (t, x, y)                          (103)

Martin Land — IARD 2008                  Relativistic Harmonic Oscillator                     June 2008     27 / 33
Harmonic Oscillator in 2+1 Dimensions
Spherical operators in number representation

Spherical operators
a± = a1 ± ia2                a± = a1 ± ia2                               (104)
[ a+ , a+ ] = [ a− , a− ] = 0            [ a+ , a− ] = [ a− , a+ ] = 2               (105)

Number operator
3  1                                3
K = N+     = ( a + a − + a − a + ) − a0 a0 +                                 (106)
2  2                                2

Casimir operator and angular momentum
Λ = N2 + N − ( a · a ) ( a · a )                                       (107)

1
L=       ( a+ a− − a− a+ )                                         (108)
2

Martin Land — IARD 2008                Relativistic Harmonic Oscillator                    June 2008     28 / 33
Harmonic Oscillator in 2+1 Dimensions
Covariant number operators on hyperspherical ground state

Hyperspherical ground state in Cartesian coordinates
1               2  2  2      i         y                   2
ψ0 (t, x, y) = N0                           e−( x +y −t )/2 e 2 arctan( x ) = ψ0 ( x, y) e t /2                  (109)
( x2   + y2 )1/4

Spherical operator a− annihilates ψ0

1√           x + iy           x + iy                               2  2  2      i         y
a − ψ0 =       2N0 −              5 +              5                      e−( x +y −t )/2 e 2 arctan( x ) = 0       (110)
4        ( x 2 + y2 ) 4   ( x 2 + y2 ) 4
Eigenstate of L
1                          1            1
L ψ0 =          ( a + a − − a − a + ) = − a − a + ψ0 = ψ0                                    (111)
2                          2            2
Timelike operator a0 annihilates ψ0
1                         2
a0 ψ0 = √ (t − ∂t ) ψ0 ( x, y) e t /2 = 0                                              (112)
2

Martin Land — IARD 2008                      Relativistic Harmonic Oscillator                         June 2008     29 / 33
Harmonic Oscillator in 2+1 Dimensions
Coherent subspaces

Commutation relations
N 0 , a ± = N 0 , a ± = N 0 , L = a0 , L = a0 , L = 0                                            (113)

[Λ, N µ ] = 0               L, N 1 = 0                  L, N 2 = 0                          (114)
States
1                                                     1
Λ |n µ ν =       µ2 −            |n µ ν                L |n µ ν =        ν+        |n µ ν               (115)
4                                                     2
Λ1                                                    Λ2

L is Hermitian and preserves n =                  n1   + n2       − n0       and   n0

L n0 n1 n2        =        i     n1 ( n2 + 1) n0 n1 − 1 n2 + 1

−i         n1 + 1 n2 n0 n1 + 1 n2 − 1                            (116)

Martin Land — IARD 2008                    Relativistic Harmonic Oscillator                           June 2008     30 / 33
Harmonic Oscillator in 2+1 Dimensions
Multiplicity of states

Label states n0 n1 n2 by n and n0

n0        n + n0 − k     k ,       k = 0, 1, ..., n + n0 ,              n0 = 1, 2, 3, ...           (117)

Multiplicity of states for given n and n0 is n + n0 + 1
Multiplicity of states for given n is inﬁnite
Deﬁne states
1          γ
ζ αβγ =          a0       ( a+ )α ( a− ) β ϕ100            α+β−γ = n                          (118)
Nγαβ
On these states
1 1                                  γ
Lζ αβγ =             ( a + a − − a − a + ) a0          ( a+ )α ( a− ) β ϕ100 = (α − β) ζ αβγ                 (119)
2 Nαβγ
Cannot satisfy the requirement for the hyperspherical solution
1        1
α − β = Λ2 = ν +           = µ+k+                       α, β, µ, k integer                   (120)
2        2

Martin Land — IARD 2008               Relativistic Harmonic Oscillator                            June 2008     31 / 33
Conclusion

3D nonrelativistic oscillator
Separate variables in Cartesian or spherical coordinates
Obtain equivalent solutions with identical eigenvalues
Unitary transformation connects the equivalent solutions

2+1 relativistic oscillator
Separate variables in Cartesian coordinates
Equivalent to covariant number representation
Timelike excitations lead to indeﬁnite spectrum or indeﬁnite norm
Suppression of timelike excitations may require ad hoc corrections
Separate variables in hyperspherical coordinates
Permits a priori suppression of timelike excitations
Solutions with positive deﬁnite spectrum and norm
Not equivalent to covariant number representation
No unitary transformation connects Cartesian and hyperspherical approaches

Martin Land — IARD 2008         Relativistic Harmonic Oscillator            June 2008   32 / 33
Thank You

Martin Land — IARD 2008    Relativistic Harmonic Oscillator   June 2008   33 / 33

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