Euclidean Relativistic Quantum Mechanics

W. N. PolyzouVic WesselsPhilip KoppTracie Michlin

The University of IowaResearch supported by theUS DOE Office of Science

July 25, 2012

Motivation

Formulate few-body relativistic quantum mechanical modelsto study few-hadron systems at the few GeV scale.

References:Phys. Rev. D85(2012)016004

Few-Body Systems, 35(2004),51

Theoretical framework

Relativistic Quantum Mechanics

• Model Hilbert space (quantum probabilities).

• Unitary representation of the Poincare group (relativisticinvariance of the quantum probabilities).

• Space-like cluster properties (required for tests ofrelativity on isolated subsystems).

• Spectral condition (required for stability of theory).

• Models motivated by field theory or effective field theory.

Euclidean formulation

• Replace 2, 3, · · · -body interactions or 4, 6, · · · -pointBethe-Salpeter kernels by 4, 6, · · · -point truncatedEuclidean Green functions.

• Formulate the model and perform all calculations entirelyin Euclidean Space - avoid any analytic continuation.

Elements of this talk

• Model assumptions, theoretical input.

• Model Hilbert space (reflection positivity?).

• The Poincare Lie algebra.

• Particles.

• Scattering (Euclidean formulation - existence? )

• Test: GeV-scale scattering using Euclidean methods.

Dynamical input

Euclidean invariant reflection positive Green functions

Gm:n(x1, · · · , xm; yn, · · · , y1)

Model Hilbert space, H

Vectors

f → (f0, f1(x), f2(x1, x2), · · · )

support of fk(x1, · · · , xk) x1, · · · , xN | x01 > x02 > x03 > · · · > x0k

Inner product

〈f |g〉 = 〈f |g〉M = (f , θGg)E =Xm,n

Zf ∗m (θx1, · · · , θxm)Gm;n(x1, · · · , xm; yn, · · · , y1)g(y1, · · · , yn)d4mxd4ny

θx = θ(x0, x) = (−x0, x)

Properties of Gm:n(x1, · · · , xm; y1, · · · , yn)

Reflection positivity

〈f |f 〉 ≥ 0

Euclidean invariance

Ex = Ox + a OtO = I

Gm:n(Ex1, · · · ,Exm;Ey1, · · · ,Eyn) = Gm:n(x1, · · · , xm; y1, · · · , yn)

Cluster property

lim|a|→∞

Gm:n(x1 + a, · · · , xk + a, xk+1, · · · , xm; y1 + a, · · · , yl + a, yl+1, · · · , yn) =

Gk:l(x1, · · · , xk ; y1, · · · , yl)Gm−k,n−l(xk+1, · · · , xm; yl+1, · · · , yn)

Reflection positivity

• Gives physical Hilbert space (with positive norm).

• Gives spectral condition (H ≥ 0).

• Satisfied by Kallen-Lehmann representation of two-pointfunctions.

• Not stable with respect to small Euclidean invariantperturbations.

• Representation theory?

Instability of reflection positivity:

G2:2(x1, x2; y2, y1) =

G1:1(x1; y1)G1:1(x2; y2)+G1:1(x1; y2)G1:1(x2; y1)+G c2:2(x1, x2; y2, y1)

Reflection positivity does not follow if G c2:2 is small and

Euclidean invariant.

G c2:2 reflection positive → G2:2 reflection positive

The solution of the Bethe-Salpeter equation with a reflectionpositive driving term and a Euclidean invariant kernel is not

automatically reflection positive.

Widder’s theorem (Bull. AMS 40(1934)321)∫f ∗(t ′)k(−t ′, t)f (t)dtdt ′ > 0 k(−t ′, t) = g(t ′ + t)

g(t) =

∫e−λtρ(λ)dλ = −i

∫p

π

e itp

λ2 + p2ρ(λ)dpdλ

G c2:2(x1, x2; y2, y1) =∫

e ip1·(x1−x2)e ip2·(x2−y2)e ip3·(y2−y1)×

g(p1, p2, p3,m2)

(p21 +m2)(p22 +m22)(p

23 +m2)

d4p1d4p2d

4p3dm2

Reflection positive, truncated, for suitable g(p1, p2, p3,m2).There is a large class of model reflection-positive Euclidean

Green functions.

Field theory vs. phenomenology

Locality

⇓

G1:3(x1; x2, x3, x4) = G2:2(x1, x2; x3, x4) = G3:1(x1, x2, x3; x4)

Not necessary for Poincare invariance, Hilbert space, orcluster properties .

Probably required for crossing symmetry?

More difficult to satisfy reflection positivity and locality.

Poincare and Euclidean invariance

X =

(t + z x − iyx + iy t − z

)X =

(iτ + z x − iyx + iy iτ − z

)

det(X ) = t2 − x2 det(X) = −(τ2 + x2)

Complex Poincare group = Complex Euclidean group

X′ = AXB X = AXB det(A) = det(B) = 1

B = A†: real Lorentz group, A,B unitary: real orthogonalgroup

Real Euclidean group

f = (f0, f1(x), f2(x1, x2), · · · ) →

fO,a := (f0, f1(O−1(x− a)), f2(O

−1(x1 − a),O−1(x2 − a)), · · · )

10 Parameter subgroup of the complex Poincare group onthe physical Hilbert space

e−βH e ia·P e iJ·n eK·n

Poincare generators

〈x|H|f〉 := 0, ∂

∂x011f1(x11),

„∂

∂x021+

∂

∂x022

«f2(x21, x22), · · ·

〈x|P|f〉 := 0,−i∂

∂ ~x11f1(x11),−i

„∂

∂ ~x21+

∂

∂ ~x22

«f2(x21, x22), · · ·

〈x|J|f〉 := 0,−i~x11 ×∂

∂~x11f1(x11),

−i

„~x21 ×

∂

∂~x21+~x22 ×

∂

∂~x22

«f2(x21, x22), · · ·

〈x|K|f〉 := 0,„~x11

∂

∂x011− x011

∂

∂~x11

«f1(x11),

„~x21

∂

∂x021− x021

∂

∂~x21+~x22

∂

∂x022− x022

∂

∂~x22

«f2(x21, x22), · · · .

Dynamics

〈x|e−βH−ia·P|f 〉 =

(f0, f1(x0 − β, x− a), f2(x

01 − β, x1 − a, x02 − β, x2 − a), · · · ) →

M2 = (∂2

∂β2+

∂

∂a· ∂∂a

)〈x|e−βH−ia·P|f 〉|β=0,a=0=

(∂2

∂β2+

∂

∂a· ∂∂a

)(f0, f1(x0 − β, x− a),

f2(x01 − β, x1 − a, x02 − β, x2 − a), · · · )|β=0,a=0

One-particle states

Orthonormal basis

|fn〉 〈fn|fm〉 = δmn

Solve for eigenstates in point spectrum of M2

〈x|(M2 − λ2)|λ〉 = 0 〈x|λ〉 =∑n

bn〈x|fn〉

∑n

〈fm|M2|fn〉bn = λ2bm

Normalizable one-particle mass eigenstate.

One-particle states

mass eigenfunction

〈x|λ〉 =∑n

〈x|fn〉bn

mass-momentum eigenfunction

〈x|λ,p〉 =∫

d3a

(2π)3/2e−ip·a〈x− a|λ〉

mass-momentum-spin eigenfunctional

〈x|λ, j ,p, µ〉 =∫SU(2)

dR

j∑ν=−j

〈x|λ,R−1p)〉D j∗µν(R)

Two Hilbert space Haag-Ruelle scattering

|Ψ±(g1, · · · gn)〉 := limt→∞

e iHtΦe−iH0t |g〉 = Ω±|g〉

〈x|Φ|g〉 =Z X Yk

〈xk |λk , jk , pk , µk〉|β=0gk(pk , µk)dpk

Cook condition - existence of Ω±Z ±∞

0

‖(HΦ− ΦH0)e−iH0t |g〉‖dt < ∞.

H0 =Xk

ωλk (pk)

Cook condition - N = 2

G4 = G2G2 + G c4

G2G2 contribution to ‖(HΦ− ΦH0)e−iH0t |g〉‖ vanishes

The Cook condition is a regularity condition on theconnected parts of Euclidean Green functions

Existence of a scattering theory does not require a local fieldtheory; only nice truncated Green functions.

Computation of scattering matrix

Sfi = limt→∞

〈g+|e iH0tΦ†e−2iHtΦe iH0t |g−〉

Kato-Birman invariance principle

H → w(H) w(H) = −e−βH β > 0

⇓

S = limn→∞

〈g+|e−ine−βH0Φ†e2ine−βH

Φe−ine−βH0 |g−〉

For fixed n, e2ine−βH

can be uniformly approximated by apolynomial in e−βH! (〈f |T (0, nβ)|g〉 = 〈f |e−nβH |g〉)

Scattering calculations in Euclidean space

Step 1: Use sharply peaked (in momentum) normalizablestates to approximate plane-wave on-shell transition matrix

elements.

〈gf |S |gi 〉 = 〈gf |gi 〉 − 2πi〈gf |δ(E+ − E−)T |gi 〉

〈p′1, µ′1, · · · ,p′n, µ′n|T |p1, µ1,p2, µ2〉 ≈〈gf |S |gi 〉 − δab〈gf |gi 〉2πi〈gf |δ(E+ − E−)|gi 〉

Solve for one-particle states

Step 2: Calculate 〈x|λ, j ,p, µ〉

〈x|λ, j〉 ≈∑n

cn〈x|fn〉

〈fn|(M2 − λ2)|λ, j〉 = 0

〈x|λ, j , g〉 :=∫

dp

j∑µ=−j

〈x|λ, j ,p, µ〉g(p, µ)

Calculate Φ

〈x|Φ|g〉 =

(∏(∂

∂β〈x− β|λ, j , gk〉|β=0

− i

∫dpωλk (p)〈x|λ, j ,p, µ〉gk(p, µ)

))

Scattering in Euclidean space

e iHt → e−ine−βH

Step 3: Replace limn→∞ by large fixed n.

〈g+|S |gi 〉

≈ 〈g+|e−ine−βHf Φ†e2ine−βH

Φe−ine−βHf |g−〉

Step 4: Uniform polynomial approximation

e2ine−βH ≈

∑cm(n)(e

−βmH)

note σ(e−βH) ∈ [0, 1] (compact)

e2inx ≈∑

cm(n)xm x → e−βH

|e2inx −∑

cm(n)xm| < ε(n) ∀x ∈ [0, 1]

⇓

‖[e2ine−βH −∑

cm(n)(e−βmH)]|ψ〉‖ < ε(n)‖|ψ〉‖

Combine all four steps

〈g+|S |g−〉 ≈

=∑

cm(n)〈g+|e−ine−βHf Φ†(e−βmH)Φe−ine−βHf |g−〉

Each approximation converges - the order of theapproximations is important (1) → (2) → (3) → (4).

Practical considerations

Is it possible to use these methods to do GeV scalescattering calculations?

First test of method: relativistic separable potential(solvable so all approximations can be tested)

M2 = 4(k2 +m2)− |g〉λ〈g |

〈k|g〉 = 1

m2π + k2

Calculate 〈k′|T (k+)|k〉 using matrix elements of e−βH innormalizable states.

Convergence with respect to wave packet width

Table 1

k0 α kw % error kw/k0[GeV] [GeV−2] [GeV]

0.1 105000 0.00308607 0.1 0.0300.3 10500 0.009759 0.1 0.0320.5 3000 0.0182574 0.1 0.0360.7 1350 0.0272166 0.1 0.0380.9 750 0.0365148 0.1 0.0401.1 475 0.0458831 0.1 0.0411.3 330 0.0550482 0.1 0.0421.5 250 0.0632456 0.1 0.0421.7 190 0.0725476 0.1 0.0421.9 150 0.0816497 0.1 0.042

Convergence with respect time “n”

Table 2: k0 = 2.0[GeV], α = 135[GeV−2]

n Re 〈φ|(Sn − I )|φ〉 Im 〈φ|(Sn − I )|φ〉50 -2.60094316473225e-6 1.94120750171791e-3100 -2.82916859895010e-6 2.35553585404449e-3150 -2.83171624670953e-6 2.37471383801820e-3200 -2.83165946257657e-6 2.37492460997990e-3250 -2.83165905312632e-6 2.37492527186858e-3300 -2.83165905257121e-6 2.37492527262432e-3350 -2.83165905190508e-6 2.37492527262493e-3400 -2.83165905234917e-6 2.37492527262540e-3

ex -2.83165905227843e-6 2.37492527259701e-3

Table 3: Parameter choices

k0 [GeV] β[GeV−1] k0 × β n

0.1 40.0 4.0 4500.3 5.0 1.5 3300.5 3.0 1.5 2050.7 1.6 1.2 2000.9 1.05 .945 1901.1 0.95 1.045 2001.3 0.85 1.105 2001.5 0.63 0.945 2001.7 0.5 0.85 2001.9 0.42 0.798 200

Table 4: Convergence with respect to Polynomial degree e inx

x n deg poly error %

0.1 200 200 3.276e+000.1 200 250 1.925e-110.1 200 300 4.903e-13

0.1 630 630 2.069e+000.1 630 680 5.015e-080.1 630 700 7.456e-11

0.5 200 200 1.627e-130.5 200 250 3.266e-13

0.5 630 580 1.430e-140.5 630 680 9.330e-13

0.9 200 200 3.276e+000.9 200 250 1.950e-110.9 200 300 9.828e-13

0.9 630 630 2.069e+000.9 630 680 5.015e-080.9 630 700 7.230e-11

Table 15: Final calculation

k0 Real T Im T % error

0.1 -2.30337e-1 -4.09325e-1 0.09560.3 -3.46973e-2 -6.97209e-3 0.09660.5 -6.44255e-3 -3.86459e-4 0.09860.7 -1.88847e-3 -4.63489e-5 0.09770.9 -7.28609e-4 -8.86653e-6 0.09821.1 -3.35731e-4 -2.30067e-6 0.09871.3 -1.74947e-4 -7.38285e-7 0.09851.5 -9.97346e-5 -2.76849e-7 0.09561.7 -6.08794e-5 -1.16909e-7 0.09641.9 -3.92110e-5 -5.42037e-8 0.0967

Calculations suggest

• Width of wave about 3% of momentum scale to get.1% accuracy in sharp-momentum S-matrix elements.

• Convergence of the polynomial approximations can betested independent of any dynamical model.

• For two-body scattering the Lehmann representation oftwo-point function solves the one-body problem.

• The scale of the time limit can be limited by choosing βnear the inverse of the energy scale.

Summary

• Model reflection-positive Euclidean Green functions can be used toformulate a relativistic quantum theory. A large class of reflectionpositive Green functions exist.

• N-body interactions are replaced by connected reflection-positive2N-point Green functions.

• Analytic continuation is not needed to compute-bound state orscattering observables.

• Existence of the S matrix can be established using Cook theorem.Yhe proof does not require locality.

• Cluster properties are easily satisfied for fixed N.

• Finite Poincare transformations on single-particle and scatteringstates can be performed.

• A test using an exactly solvable model suggests that GeV scalescattering cross sections can be accurately computed using Euclideanmethods,

Future studies

• Solutions of the one-body problem.

• NN scattering - G4 → G2G2 + G c4 .

• Euclidean Nakanishi representation and reflectionpositivity?

• Gauge theories - reflection positivity only for colorsinglets.

• Current matrix elements

Thanks!

Conference organizers and staff and students

Approximation 4: Use Chebyshev polynomials

f (x) ≈ 1

2c0T0(x) +

N∑k=1

ckTk(x)

cj =2

N + 1

N∑k=1

f (cos(2k − 1

N + 1

π

2)) cos(j

2k − 1

N + 1

π

2)

f (e−βH) ≈ 1

2c0T0(e

−βH) +N∑

k=1

ckTk(e−βH)

f (x) = e2inx

|e2inx − PN(x)| < 2nN+1

(N + 1)!

One-parameter groups and semigroups

Euclidean time translations - contractive Hermitiansemigroup

f (x) → f (x− (β, 0, 0, 0)) e−βH β > 0

Rotations and space translations - unitary one-parametergroups

f (x) → f (x− (0, a)) e ia·P f (x) → f (R−1x) e iJ·nψ

Rotations in space-time planes - local symmetric semigroups

f (x) → f (R−1x) eK·nψ

Generators H,P, J,K are self-adjoint on H and satisfyPoincare commutation relations

Finite Poincare transformationsof one-particle states (λ ∈ σpp)

One-particle subspaces are irreducible subspaceswith respect to the Poincare group

⇓

〈x|U[Λ, a]|λ, j , p, µ〉 =

jXµ′=−j

Zdp′〈x|λ, j , p′, µ′〉Dλ,j

p′µ′;p,µ[Λ, a]

Dλ,jp′µ′;p,µ[Λ, a] = 〈λ, j , p′, µ′|U[Λ, a]|λ, j , p, µ〉

The mass λ spin j irreducible representation of the Poincare group areknown.

Finite Poincare transformationsof scattering states

Ω±|g〉 = limt→±∞

e iHtΦe−iH0t |g〉

Acceptable wave operators satisfy

U[Λ, a]Ω± = Ω±Uf [Λ, a]

Uf [Λ, a] = ⊗Uλk ,jk [Λ, a]

U[Λ, a]Ω±|g〉 = Ω±Uf [Λ, a]|g〉