Cluster Separability In RelativisticQuantum Mechanics

W. N. Polyzou∗ - The University of IowaB. D. Keister - NSF

Phys. Rev. C86(2012)014002∗ Research supported by the in part by

the US DOE Office of Science

August 21, 2012

Outline of talk

• Overview

• The Bakamjian-Thomas construction

• Cluster properties

• The Sokolov construction

• Test model

• Results and conclusion

Physical requirements for relativistic QM models

• Hilbert space (quantum probability amplitude = Hilbertspace inner product)

• Unitary representation of the Poincare group (soprobabilities have same values in all inertial frames)

• Cluster separability (justifies local tests of specialrelativity)

• Spectral condition (stability of theory)

Observations

• All of these properties can be realized in Poincareinvariant quantum mechanics of a fixed number ofparticles.

• Poincare invariance or cluster properties can be easilyrealized separately.

• Satisfying both requires a complicated recursiveconstruction.

• The size of the corrections that restore clusterproperties have never been quantitatively investigated.

History

• Wigner 1939: Relativistic quantum mechanics = unitaryrepresentation of the inhomogeneous Lorentz (Poincare)group.

• Dirac 1949: Poincare commutation relations putnon-linear dynamical constraints on the generators.

• Bakamjian Thomas 1951: N = 2 add interactions to themass keeping the spin free solves the dynamicalconstraints.

• Coester 1965: N = 3 cluster properties of the S-matrix,not generators.

• Sokolov 1971: General solution.

Bakamjian-Thomas construction

|(M, j)P, µ; l , s〉 =∑∫

dp1dp2|(m1, j1)p1, µ1〉 ⊗ |(m2, j2)p2, µ2〉×

〈(m1, j1)p1, µ1(m2, j2)p2, µ2|(M, j)P, µ; l , s〉︸ ︷︷ ︸Poincare group Clebsch-Gordan coeff.

MI = M0 + V

〈(M, j)P, µ; l , s|V |(M ′, j ′)P′, µ′; l ′, s ′〉 =

δ(P− P′)δjj ′δµµ′〈M, l , s‖v j‖M ′, l ′, s ′〉

Find complete a set of eigenstates of MI in the free-particleirreducible basis

〈(M, j)P, µ; l , s|(λ, j ′)P′, µ′〉 = δ(P− P′)δjj ′δµµ′φλ,j(M, l , s)

(λ−M)φλ,j(M, l , s) =∑∫ ′

dM ′〈M, l , s‖v j‖M ′, l ′, s ′〉φλ,j(M, l , s)

U(Λ, a)|(λ, j)P, µ〉

∑ν

|(λ, j)ΛΛΛP, ν〉e iΛP·a√ωλ(ΛP)

ωλ(P)D jνµ(Rw (Λ,P))

Poincare generators

U(Λ, a) = e i∑

giGi

Gi = −i ∂∂gi

U(Λ, a)|g=0

Gi ∈ {H,P, J,K}

Cluster properties

H =∑i

Hi +∑ij

Hij +∑jk

Hijk + · · ·

P =∑i

Pi +∑ij

Pij +∑ijk

Pijk + · · ·

J =∑i

Ji +∑ij

Jij +∑ijk

Jijk + · · ·

K =∑i

Ki +∑ij

Kij +∑ijk

Kijk + · · ·

Iµ(x) =∑i

Iµi (x) +∑ij

Iµij (x) +∑jk

Iµijk(x) + · · ·

The problem

Instant form caseInteractions in K; no interactions in J

[K i ,K j ] = −iεijkJk

[K i12,K

j32]

is a 3-body operator that must be canceled

Sokolov constructionThree particles, particles 1 and 2 interact

Two S-matrix equivalent treatments

〈P′, j ′3, µ′3, p′3, k ′, j ′, l ′, s ′, µ′|V TP12 |P, j ′3, µ′3, p3, k, j , l , s, µ〉 =

δ(P′ − P)δj′3 j3δµ′3µ3δ(p′3 − p3)δj′jδµ′µ〈k ′, l ′, s ′‖v j

12‖k, l , s〉

q3 := ΛΛΛ−1(P/M(k))p3 j = Rw (P, p12)j j3 = Rw (P, p3)j3

〈P′, j3, µ3, q′3, k′, j ′, l ′, s ′, µ′|V BT

12 |P, j3, µ3, q3, k, j , l , s, µ〉 =

δ(P′ − P)δj′3 j3δµ′3µ3δ(q′3 − q3)δj′jδµ′µ〈k ′, l ′, s ′‖v j

12‖k, l , s〉

⇓

⇓

UTPij ,k (Λ, a) = Uij(Λ, a)⊗ Uk(Λ, a) UBT

ij ,k (Λ, a)

⇓

STPij ,k = SBT

ij ,k , 〈k ′, l ′, s ′‖S j12‖k , l , s〉

⇓

∃Aij ,k unitary

Aij ,kUTPij ,k (Λ, a)A†ij ,k = UBT

ij ,k (Λ, a)

⇓

MBT := MBT12,3 + MBT

23,1 + MBT31,2 − 2MBT

0 jBT := j0

⇓

UBT (Λ, a)

⇓

U(Λ, a) := A†UBT (Λ, a)A

A = eln(A12,3)+ln(A23,1)+ln(A31,2)

M :=

A†(A12,3M12,3A

†12,3 + A23,1M23,1A

†23,1 + A31,2M31,2A

†31,2 − 2MBT

0

)A

⇓

H =∑i

Hi +∑ij

Hij + H123

P =∑i

Pi +∑ij

Pij + P123

J =∑i

Ji +∑ij

Jij + J123

K =∑i

Ki +∑ij

Kij + K123

• The operators Aij ,k only generate three-bodyinteractions, H123,P123, J123,K123.

• The three-body interactions are needed to restore thecommutation relations.

• The operators Aij ,k are functions of the two-bodyinteractions.

• The one and two-body parts of the generators aredetermined by cluster properties.

• N = 3 is a special case where S = SBT . This is not truefor N > 3.

• Aij ,k → I implies U(Λ, a)→ UBT (Λ, a).

Test four-body model

• Electron scattering from nucleon 3; nucleons 1 and 2 arebound.

• The bound pair does not interact with nucleon 3 or theelectron.

• Nucleons are treated as spinless; the current is replacedby a scalar density. The dynamics is S-matrix equivalentto a Malfliet-Tjon S-matrix.

• The three-nucleon system is modeled using the BTmethod or the scattering equivalent tensor productmethod.

Current:

p3 p12

p′3 p′12

q

〈12⊗ 3|j(0)|12⊗ 3′〉 = 〈(12, 3)BT |A12,3j(0)A†12,3|(12, 3)′BT 〉

Differences in〈12⊗ 3|j(0)|12⊗ 3′〉

and〈(12, 3)BT |j(0)|(12, 3)′BT 〉

are due to the operator A12,3.

Compare∫dp′12TP〈p3,p12, φ|j(0)|p′3,p′12, φ〉TP

with∫dp′12BT 〈p3,p12, φ|j(0)|p′3,p′12, φ〉BT

as a function of p12.

Plots show

F (p′3 − p3, p12) :=∫dp′12TP〈p3, p12, φ|j(0)|p′3, p′12, φ〉TP −

∫dp′12BT 〈p3, p12, φ|j(0)|p′3, p′12, φ〉BT∫

dp′12TP〈p3, p12, φ|j(0)|p′3, p′12, φ〉TP +∫dp′12BT 〈p3, p12, φ|j(0)|p′3, p′12, φ〉BT

under different kinematic conditions

Cluster propertites: the results should be independent of p12!

Instant form perp:

instant form (BT-TP)/TP vs. q, P12; q perp to P12

-10-5

0 5

10q [fm-1] -6-4

-2 0

2 4

6

P12 [fm-1]

0.0 x 100

5.0 x 10-4

1.0 x 10-3

Instant form parallel:

instant form (BT-TP)/TP vs. q, P12; q parallel to P12

-10-5

0 5

10q [fm-1] -6-4

-2 0

2 4

6

P12 [fm-1]

0.0 x 100

5.0 x 10-4

1.0 x 10-3

1.5 x 10-3

Front form perp:

front form (BT-TP)/TP vs. q, P12; q perp to P12

-10-5

0 5

10q [fm-1] -6-4

-2 0

2 4

6

P12 [fm-1]

0.0 x 1001.0 x 10-32.0 x 10-33.0 x 10-34.0 x 10-35.0 x 10-3

Front form par:

front form (BT-TP)/TP vs. q, P12; q parallel to P12

-10-5

0 5

10q [fm-1] -6-4

-2 0

2 4

6

P12 [fm-1]

0.0 x 1001.0 x 10-32.0 x 10-33.0 x 10-34.0 x 10-35.0 x 10-3

Point form perp:

point form - (BT-TP)/TP vs. q, P12; q perp to P12

-10-5

0 5

10q [fm-1] -6-4

-2 0

2 4

6

P12 [fm-1]

0.0 x 100

2.0 x 10-5

4.0 x 10-5

6.0 x 10-5

Point form parallel:

point form - (BT-TP)/TP vs. q, P12; q parallel to P12

-10-5

0 5

10q [fm-1] -6-4

-2 0

2 4

6

P12 [fm-1]

-1.0 x 10-30.0 x 1001.0 x 10-32.0 x 10-33.0 x 10-3

vary k0, parallel:

1 x 10-4

1 x 10-3

1 x 10-2

1 x 10-1

0 0.5 1 1.5 2

|(B

T-TP)/

TP|

k0 [fm-1]

all forms - (BT-TP)/TP vs. k0, q = 10, P12 parallel=5

point parallelinstant parallel

front parallel

vary k0, perp:

1 x 10-7

1 x 10-6

1 x 10-5

1 x 10-4

1 x 10-3

1 x 10-2

1 x 10-1

0 0.5 1 1.5 2

|(B

T-TP)/

TP|

k0 [fm-1]

all forms - (BT-TP)/TP vs. k0, q = 10, P12 perp=5

point perpinstant perp

front perp

vary be, perp:

1 x 10-5

1 x 10-4

1 x 10-3

1 x 10-2

0 2 4 6 8 10

|(B

T-TP)/

TP|

BE [MeV]

all forms - (BT-TP)/TP vs. BE, q = 10, P12 perp=5

point perpinstant perp

front perp

vary be, parallel:

1 x 10-3

1 x 10-2

0 2 4 6 8 10

|(B

T-TP)/

TP|

BE [MeV]

all forms - (BT-TP)/TP vs. BE, q = 10, P12 parallel=5

point parallelinstant parallel

front parallel

vary k0, quark perp:

1 x 10-5

1 x 10-4

1 x 10-3

1 x 10-2

1 x 10-1

1 x 100

0 1 2 3 4 5

|(B

T-TP)/

TP|

k0 [fm-1]

all forms - |(BT-TP)/TP| vs. k0, q = 10, P12 perp=5

instant perpfront perppoint perp

vary k0, quark parallel:

1 x 10-5

1 x 10-4

1 x 10-3

1 x 10-2

1 x 10-1

1 x 100

0 1 2 3 4 5

|(B

T-TP)/

TP|

k0 [fm-1]

all forms - |(BT-TP)/TP| vs. k0, q = 10, P12 parallel=5

instant parfront parpoint par

vary be, quark perp:

1 x 10-6

1 x 10-5

1 x 10-4

1 x 10-3

1 x 10-2

1 x 10-1

200 250 300 350 400 450 500 550 600

|(B

T-TP)/

TP|

diquark mass [MeV]

all forms - |(BT-TP)/TP| vs. md, q = 10, P12 perp=5

instant perpfront perp

point B perp

vary be, quark parallel:

1 x 10-7

1 x 10-6

1 x 10-5

1 x 10-4

1 x 10-3

1 x 10-2

1 x 10-1

1 x 100

200 250 300 350 400 450 500 550 600

|(B

T-TP)/

TP|

diquark mass [MeV]

all forms - |(BT-TP)/TP| vs. md, q = 10, P12 perp=5

instant perpfront perppoint perp

Conclusion

• For problems with nuclear physics scales the operatorsAij ,k that restore cluster properties can be neglected.

• Exchange current generated by Aij ,k can be neglected innuclear physics applications.

• There are cluster problems in all of Dirac’s forms ofdynamics. The scales of the cluster errors arecomparable.

• For systems of hadrons with sub-nucleon degrees offreedom the corrections are larger; but still would bedifficult to observe.