Discrete-time quantum mechanics. III. Spin systems

PHYSICAL REVIEW D VOLUME 35, NUMBER 10 15 MAY 1987

Discrete-time quantum mechanics. III. Spin systems

Carl M. BenderDepartment of Physics, Washington University, St. Louis, Missouri 63130

Fred CooperTheoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

Kimball A. MiltonDepartment of Physics and Astronomy, Uniuersity of Oklahoma, Norman, Oklahoma 73019

Stephen S. PinskyDepartment of Physics, Ohio State Uniuersity, Columbus, Ohio 43210

L. M. Simmons, Jr.Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545

(Received 10 November 1986)

A program is underway to obtain numerical solutions to quantum field theories by formulatingthem in terms of operator difference equations on a Minkowski space-time lattice. The most crucialunsolved problem is implementing non-Abelian gauge invariance. This paper initiates the study ofthis difficult problem by treating spin systems. The central problem here is to preserve exactly the(q-numberj non-Abelian commutation relations at each lattice site. The solution we propose requiresthat the spin variables be expressed in terms of more fundamental oscillator variables which satisfythe Heisenberg algebra.

I. INTRODUCTION

In contrast with the conventional Monte Carlo ap-proach to lattice gauge theory, we have been pursuing aprogram in which the lattice plays a fundamentalrole. ' " Rather than using the lattice as a computationaldevice to regulate a continuum theory, we use the methodof finite elements to construct ab initio on a Minkowskispace-time lattice a fully consistent, unitary quantumtheory. That is, we formulate a set of operator difference(rather than differential) equations that exactly preserveequal-time commutation relations at the lattice sites. Themain advantage of this formulation is that these operatordifference equations can be solved to obtain very accurateand nonperturbative numerical approximations to matrixelements and to the energy spectrum. In addition, in pre-vious papers we have shown that this approach solves theproblem of fermion doubling and readily accommodatesAbelian gauge invariance.

The main unsolved problem in this program is the im-plementation of non-Abelian gauge invariance. Clearly,the introduction of a lattice breaks the space-time sym-metries such as rotational and Lorentz invariance. How-ever, in the limit as the lattice spacing goes to zero, thesesymmetries are unambiguously restored. ' In contrast, ifthe introduction of a lattice regulator breaks an internalcontinuous local gauge symmetry, the symmetry mayremain permanently broken even in the limit of zero lat-tice spacing. Our goal, therefore, is to find a way to con-struct lattice operator difference equations that are mani-festly gauge invariant. While we were able to accomplish

H= +V(q),2

(1.2)

this in a straightforward manner for the Abelian gauge in-variance of quantum electrodynamics, non-Abeliantheories present a much more difficult challenge. Theessential difficulty is easy to identify: commutators ofnon-Abelian quantum variables are operators rather thanc numbers. It is hard to find a way to preserve exactlyequal-time commutation relations that are operators.

In this paper, we begin the study of non-Abelian sys-tems by considering spin Hamiltonians. We considerHamiltonians of the form H(S), where S(t) is a three-component vector whose components at equal times satis-fy the commutation relations of the group SU(2):

[S;(t),SJ(t)]=i e0kSk(t) .

The novelty of this paper is that we are able to preserve(1.1) exactly on a time lattice by expressing the operator Sas a product of more fundamental oscillator variables.These oscillator variables satisfy equal-time c-number(rather than q-number) Heisenberg commutation relationsat the lattice sites. The procedure described here is com-pletely general: it applies to any number of spin variables,and generalizes when the SU(2) algebra is replaced by thealgebra of SU(X).

When we express the spin variables in terms of oscilla-tor variables we obtain Hamiltonians that are of a dis-tinctly more complex form than the Hamiltonians con-sidered in the previous two papers in this series. In I weconsidered Hamiltonians of the form

35 3081 1987 The American Physical Society

3082 BENDER, COOPER, MILTON, PINSKY, AND SIMMONS 35

where the canonical position and momentum operatorpair (q,p) constitute one degree of freedom, and the po-tential V(q) is a function of the position variable only. InII v e considered systems having two degrees of freedom( q,p) and ( r, k) described by a Hamiltonian

p2 f 2

H= + +V(q r),2 2

(1.3)

whose potential V(q, r) is still a function of the positionvariables only. In order to treat spin systems in this pa-per, we are led to completely general Hamiltonians of theform

H =H(q, p;r, k) . (1.4)

In (1.4) the coordinate variables are not separated fromthe momentum variables as they are in (1.2) and (1.3).Consequently, entirely new technical problems arise in-volving operator ordering.

We considered the question of how to deal with such ageneral Hamiltonian H(q, p) with one degree of freedomin a previous publication. ' We expect an enormous in-crease in difficulty in going from one to many degrees offreedom in the case of a general Hamiltonian. Such an in-crease in complexity for the Hamiltonians of the form(1.3) was found in going from paper I to paper II of thisseries. The additional subtlety is connected with the prob-lem of showing that the degrees of freedom that are in-dependent at the initial time remain independent at subse-quent lattice sites.

We have tried to organize this paper to explain as clear-ly as possible the necessity of replacing variables thatsatisfy q-number commutation relations by more funda-mental variables that satisfy c-number commutation rela-tions. In Sec. II we consider a very simple spin Hamil-tonian

H =S Sy+SyS (1.5)

When we discretize the resulting Hamilton's equations us-

ing the finite-element prescription, we obtain a set ofoperator difference equations which do not preserve thecommutation relations (1.1). (This failure occurs at orderh, where h is the lattice spacing. ) This is more than alittle surprising in view of the simplicity of the model be-

ing considered. Moreover, no simple reordering of thespin operators in the continuum Hamiltonian like thatconsidered in Ref. 10 eliminates this failure of unitarity.This elementary model clearly demonstrates the difficultyof discretizing Lie-group variables.

In Sec. III we consider an even simpler class of spinHamiltonians, which, as far as we know, is the only classfor which the straightforward finite-element prescriptionworks directly. These are the Hamiltonians that dependonly on one spin variable, such as

[c,c ]=[d,d ]=1, [c,d ]=[c,d]=0. (1.8)

Similar representations exist for any SU(N) group. '

When the spin Hamiltonian is rewritten in terms of theoscillator variables c,d and totally symmetrized in thesevariables, following the procedure of Ref. 10, the resultingoperator difference equations preserve unitarity. We showin Sec. IV that the failure of unitarity no longer occurs forthe model of (1.5).

It is essential to understand why the procedure of re-placing spin operators by creation and annihilation opera-tors, and then totally symmetrizing in these variables,yields a discrete theory that satisfies the requirements ofunitarity. For this purpose we develop in Sec. V an exten-sive formal operator machinery, which was introduced inSec. III, and which can be used in proofs of unitarity.The actual proof of unitarity is indicated in Sec. VI.

In the Appendix we consider the possibility of con-structing classes of spin Hamiltonians whose correspond-ing lattice equations of motion are unitary. We constructthese Hamiltonians as products of Hahn polynomials inthe individual oscillator variables following the work inRef. 10. The resulting Hamiltonians emerge as new in-teresting polynomials in S,. The properties of these poly-nomials are discussed in some detail.

II. FAILURE OF LATTICE UNITARITYFOR ~:S&Sy +Sy Sz

In this section we show that even in the simplest ofHamiltonians, a naive application of the method of finiteelements will not yield a unitary lattice theory if the quan-tum variables obey a Lie algebra. We consider the Hamil-tonian

H =S Sy+SyS

The continuum equations of motion are

(2. 1)

S =S S,+S,S„,

structing an explicit closed-form expression for the uni-tary time-evolution operator (the transfer matrix). TheAppendix develops polynomial properties of Hamiltoniansof the type (1.6).

A powerful and completely general way to discretizethe field equations arising from arbitrary spin Hamiltoni-ans such as that in (1.5) is based on the Schwinger repre-sentation for SU(2) spin operators 3

S= —,' a'o.a,

where the a = (c,d) is a two-component annihilationoperator. The components of a satisfy the commutationrelations

H =H(S, ) . (1.6) Sy Sy S S Sy (2.2)

For Hamiitonians of the form (1.6) we prove unitarityfrom the discrete equations of motion. This proof re-quires the use of some formal operator techniques whichare further developed in Sec. V. We then put the proof ofunitarity on a much more explicit footing by actually con-

S =2(SX —S ) .

The linear finite-element prescription for discretizing con-tinuum equations of motion replaces undifferentiatedvariables by averages and first derivatives by forwarddifferences. Here we consider two adjacent lattice sites

35 DISCRETE-TIME QUANTUM MECHANICS. III. SPIN SYSTEMS 3083

which we label 1 and 2. To simplify notation, wehenceforward represent the spin variable S„at lattice sites1 and 2 by x1 and x2. Similarly we use the notationV1,V2 and z1,z2 to represent S~ and S, at the sites 1,2. Interms of the x V, and z variables, the finite elementdiscretization of (2.2) is

+i (x] —y] )h + (2.7)

We can now compute the commutator of x2 with y2 toorder h:

[X2 3 2] = ]zz+ 41'(y]z] 'y] —x]z] 'x] )h

X2 —X1

V2 —V1

h

Z2 Z1

X2+X12

Z2+Z1

2

V2+V12

Z2+Zl2

V2+V12

Z2 +Z]2

X2+X12

Z2 +Z12

V2+3'1

2X2+X1

2

2

(2.3a)

(2.3b)

(2.3c)H =axy+a*yx+P(y —x )+yz . (2.8)

Observe that the difference between [x2,y2] and iz2 is notzero but rather of order h, showing that the lattice equa-tions (2.3) violate unitarity. It is not surprising that theviolation first occurs in order h, because we alreadyknow that for any differential equation the expansion (2.5)agrees with the continuum result through order h, andthat the continuum equations, of course, do not violateunitarity.

We do not know of any simple way to avoid this viola-tion of unitarity. It is natural, in the spirit of Ref. 10, toseek an orthogonal rotation of the spin variables in (2.1)that will lead to lattice equations consistent with unitari-ty. The physical content of such a Hamiltonian would beunchanged. An example of such a rotated Hamiltonian is

where h is the lattice spacing. The statement of unitarityis that the SU(2) commutation relations hold at each lat-tice site. That is, if x1, y1, and z1 satisfy

It is a straightforward calculation to show that there is nochoice of the parameters a, P, and y for which the associ-ated lattice equations preserve unitarity.

[x],y]] =iz], [y],z]]=ix], [z],x]]=iy], (2.4)

x, =x, +Ah +Bh'+Ch'+

y2 ——y1+Dh +Eh +Fh +z2 ——z]+Gh +Hh +Ih +

(2.5)

We insert (2.5) into (2.3) and compare powers of h to ob-tain explicit solutions for the operator coefficientsA, B,C, . . . . The results are

A =X1Z1+Z1X13 3B =2y1x 1V1+2z1x]z1—2x1 + —,x1,

C = Z] X]Z] +Z]X]Z] —4(X] Z]X] +X]Z]X] )2 2 2 2

3(X]Z] +Z']X] )

(2.6a)

(2.6b)

(2.6c)

D = —V1Z1 —Z1V1

3 3E =2z1y1z1+2x]V1x1 —2V1 + —,V]

F = —z1 V1z1+z1y]z1 +4V1 z]V1+ 4V]z1y2 2 2 2

+ —.'

(3 1z1 +Z 131»

(2.6d)

(2.6e)

(2.6f)

then as a consequence of (2.3), xq, y2, and zz satisfy thesame equations.

It is easy to see that Eqs. (2.3) violate unitarity. Wemerely solve (2.3) perturbatively in powers of the latticespacing h. We seek solutions for x2, V2, and z2 in theform

III. SPIN HAMILTONIANS OF THE FORM H (z)

From the example considered in Sec. II it is apparentthat a lattice treatment of spin systems requires a moresubtle approach than that used in our previous work.However, there is a class of spin Hamiltonians for whichthe straightforward finite-element approach is successful.

Hamiltonians of the form

H =H(z), (3.1)

in which only one of the spin variables appears, are amen-able to a conventional finite-element lattice discretization.The equations of motion for the system described by (3.1)are

x = i [x,H (—z)],y= —1[y H(z)l,z=0.

(3.2)

It is convenient at this point to introduce a new nota-tion in operator equations that contain commutators. Weuse the symbol D„,f (x,y, z, . . . ) to mean the sum of allterms each of which is formed by replacing one and onlyone of the operators x in f by a, where a,x,y, z, . . . mayall be noncommuting operators. The operator D, is onlydefined when applied to expressions f that are polynomi-als in the operator x. Here are several examples:

Dx ax =xa +ax,G =2(y] —x] ),H = —4y1z 1V1 —4x]z]x1 —2z1,

I = 4(x] —3'1 )+63',z, 3'] —6x,z, x,4 4 2 2

+2~(y] —x] ).

(2.6g)

(2.6h)

(2.6i)

D,xVxy =aVxV +xVaV,

D„,yz =0 .

Note that D,f is just Bf/Bx, in the usual sense of aderivative with respect to an operator. We postpone a


more complete discussion of the properties of the substitu-tion operator D, for Sec. V; here, we list the most ele-mentary properties, which we use in this section. Thelinearity of D implies that

Dx, a+b =Dx, a +Dx, b»~

Dx, aa Dx, a

D...(f +g) =D..f+D.,.gD„,af =aD„,f,

(3.3a)

(3.3b)

(3.3c)

(3.3d)

where cx is a c number.We illustrate the connection between the operator D

and commutators by reexpressing the continuum equa-tions of motion (3.2) in terms of D:

h[z),x l=iy) ——D, , —ix +(h/2)D H(z, )H(z) ) .

1' 1 z&, tz&, x](3.9)

Comparing (3.9) and (3.8b), we observe that [z),x] and iysatisfy the same equation.

We now argue that there is a unique solution to (3.8b)that remains finite as h goes to zero. In fact this solutioncan be explicitly constructed as a series in powers of h.The coefficients are unique because they are determinediteratively. (We do not consider here the convergence ofthe series. ) Therefore we conclude that [z, ,x] =iy Th.usfrom (3.6) it follows that [z2,x2] =iy2.

By an identical argument in which the roles of x and yare interchanged, we prove that

[z2~y2] lx2

x = D, yH—(z),

y =D, „H(z),

The finite-element lattice transcription of (3.4) is

(3.4)

Finally, we show that [x2,y2]=iz2 W.e do this bycommuting (3.5a) with y and x with (3.5b). Adding theresults and using [z),x]=iy and [z),y] = ix we o—btain

2

X2 —X)D, H(z)—, (3.5a)

=iD, „D, H(z) ) iD, ~D—, „H(z) )=0 . (3.10)

y2 —y& =D, ,H(z),

ZQ Zf

h=0,

(3.5b)

(3.5c)

This completes a formal proof of unitarity.

B. Explicit construction of the latticetime-evolution operator

where x =(x, +x2)/2, y =(y) +y2)/2, and z =(z)+ z2)/2.

A. Proof of unitarity

Z2 —Z —Z] o (3.6)

This is only true for Hamiltonians of the form (3.1).Thus, only the two operators x and y evolve in time.

We begin by proving that [z2,x2] =iy 2 Equations.(3.5a) and (3.5b) can be rewritten as

In this subsection we give a formal argument that thelattice equations (3.5) preserve the SU(2) commutation re-lations.

As we will see, the crucial reason for the unitarity of(3.5) is that, as a consequence of (3.5c),

For especially simple Hamiltonians of the class (3.1) itis easy to construct explicit expressions for the latticetime-evolution operator U, which has the property that itadvances the dynamical operators one time step:

Ux& U =xz,—]

—1

Uz& U '=z2 .

(3.1 1)

Xp —X) yI +ye2

The reason that this explicit construction is feasible is thatU is a function of z& only. As we will see, the expressionfor U is not unique (a condition it shares with the contin-uum transfer matrix).

Example 1: H(z) =z. For this Hamiltonian, thediscrete equations of motion are

x=x) — D, H(z, ), —h(3.7a)

hy=y, + —D, „H(z, ) . (3.7b)

h) z(,y(+(hl2)D H(z() ) (3.8a)

hy =y) +—Dz, z —(hl2)D H( )H(z() .

z, ,y

Next, we commute (3.8a) with z, :

(3.8b)

We eliminate y from (3.7a) using (3.7b) and similarlyeliminate x from (3.7b) using (3.7a):

h

X)+X22

Zp Z]

where

Zp Z] =0h

The solution to these equations is

x2 =x ) coscx —y1 sino!

yz ——y& cosa+x&sina,

(3.12)

(3.13)

DISCRETE-TIME QUANTUM MECHANICS. III. SPIN SYSTEMS 3085

4htan+ =

4 —h(3.14)

We can rewrite these solutions in alternate forms by use ofthe identities

We write

I'8(a j (3.15)

g(z)d) ——d)g(z+ —, ),g(z)c, =c]g(z ——, ),

(3.26)

and differentiate (3.11) with respect to a to find differen-tial equations for 0:

where g(z) is an arbitrary function of z. The proof of un-itary is now immediate because, from (3.25) and (3.26),

d2 ——deaf '(z), (3.27)

(3.16) and f defined in (3.24) is a pure phase.The unitary time-evolution operator U(z) is given by

A solution to (3.16) is

0'=zi,from which we obtain

t9=az, .

(3.17)

(3.18)

cp ——U(z)e, U (z)=, c, ,U(z)

U(z+ —, )

where we use (3.26). Comparing with (3.24) we get

U(z) f( )U(z+ —,

')

(3.28)

(3.29)

Note that the solution for 0' in (3.17) is ambiguous up toaddition of any function of the Casimir operator,x +y +z . Further a in (3.18) is ambiguous up to an

I 2.7TZ ladditive integer multiple of 2m. . (Note that e com-mutes with x &, y&, and z&.)

Example 2: H(z)=4z. For this example it is mostconvenient to use the oscillator representation (1.7). Ex-plicitly the Schwinger representation for the spin variables1s

Since U is a unitary operator, it can be written as

U ( ) e iiL(z)

From (3.29) we conclude

1 ih (4—z+ 1)/21+ih (4z+ 1)/2

which may be summed to give the general result

(3.30)

(3.31)

x = —,(c d+d c),

y= —(d c —c d),2

z= —,(c c —d d) .1

(3.19)

X+1+1 —h (2z+n + —, )

(z)+i ln Q,

0 1+h(2z+n+ —, )

(3.32)

The Hamiltonian can be rewritten in these variables as

H =4z =c cc c +d dd d —2c cd d . (3.20)

The resulting equations of motion for the oscillator vari-ables are

c= i (4z+—1)c,

d = —i ( —4z -+ 1)d,from which it follows that

z=O .

(3.21a)

(3.21b)

(3.22)

1 ih (4z+1)/2—C2= c, —:(z)ci .

1+ih (4z+ 1)/2(3.24)

Similarly,

When we put (3.21a) on the lattice we obtain

ihc2 —c ~

————(4z + 1)(cq+c ~ ),2

(3.23)

where, because the operator z is constant, we havedispensed with the representation (3.19) for z. Equation(3.23) may be immediately solved for cz..

r IV. THE UNITARY DISCRETIZATIONOF H =S„Sy+SyS„

In this section we return to the Hamiltonian

H =3(xy +yx) (4.1)

The above construction of the unitary time-evolutionoperator is ambiguous because A,(z) can be arbitrarily de-fined on the interval 0&z & —, . (The eigenvalues of z, ofcourse, are half-integers. )

It is straightforward, although somewhat tedious, toshow, from (3.32), that as h ~0, A, (z) ~4hz . We thus re-cover the continuum limit, U(z) =e'"

With the class of Hamiltonians H(z) that depend onlyon a single spin variable z =S, there is associated an in-teresting set of polynomials W „(z). These polynomialsplay a role analogous to that played by the continuousHahn polynomials for the case of Hamiltonians H(q, p)that are arbitrary functions of the canonical position andmomentum variables q and p (Ref. 10). We discuss someof the properties of the polynomials W „(z) in the Ap-pendix.

1 —ih ( —4z+ 1)/21+ih ( —4z+1)/2

(3.25) discussed in Sec. II. To treat this Hamiltonian properlyon the lattice, we will follow the strategy proposed in Ref.

BENDER, COOPER, MILTON, PINSKY, AND SIMMONS 35

10. There it was shown that it is necessary to symmetrizefully the quantum variables in the Hamiltonian in orderto obtain lattice difference equations that are consistentwith unitarity. However, in the case of spin Hamiltoni-ans, a total symmetrization of the spin variables is not in

general sufficient to ensure unitarity. Rather, we intro-duce the oscillator representation (1.7), and totally sym-metrize the Hamiltonian in terms of the oscillator vari-ables in (3.19).

In terms of the variables c and d the fully symmetrizedversion of (4. 1) is

H =—[ccd dt+cd cd +d ccd +d"cd c+d d cc4

+cd d c —(ddc c +dc dc +c"ddc +c dc d

c~(c, +c2 )/2—:c (4 4)

and so on. The average variables c and d do not com-rnute.

The finite-element transcription of (4.3) is

C2 —C) = —(ddc +dc d+c dd),

h=ccd +cd c+d cc,

= —(8'd'c+d'cd'+cd'd'),(4.5)

continuum c and d commute, so that it may seem silly towrite three equivalent terms on the right sides of (4.3).However, in diseretizing (4.3) we replace each operator onthe right side by an average at adjacent lattice sites:

+c c dd +dc c"d)],

and the continuum equations of motion are

c= —(ddct+dc d+c dd),

d =ccd +cd c +d cc,= —(ddc+dcd +cdd ),

(4.2)

(4.3) c2 ——c~+Ah+Bh +Ch + - . .

d2 ——d)+Dh +Eh +Fh(4.6)

h=c c d+c dc +dc c

As in Sec. II, we solve (4.5) by expanding cq and dq inpowers of h:

d =ccd+cdc +decBecause we have totally symmetrized the Hamiltonian,the right sides of (4.3) are also totally symmetric. In the

Vr'e determine the operator-valued coefficients A, B,C, . . .by substituting (4.6) into (4.5) and comparing powers of h.The results are

A = —3d, c~, B = ——,(2c, d~c, di —c,d, d, —c, c, ),2 t 9 2 f t 2 t2 2

C= ——, ( —6c~d, c, d~+2c~ d~ c~ +c~ c,d~ —2c, d, d~ +d~ cid~ +2d, c,d, +3c,d, c, +c, d, —d~ c~),3 ~2 ~ 2 2 f3 4 ~ t2 3 f3 4 t f2 3 2 t2 3 T2 2

9 2i f 2 t2 2(4.7)

D =3c, d, , E = ——,(2c~d, c~d, —c~ d~c, —d& d, ),F= —

4 (6c~ dtctd~ —2c, d~ d, —d, c, d, +2c,d, c, —c, c, d, —2c, c,d, —3c& d, d, —d~ ci +c, d~).27 3 t t2 2 2 f3 4 ~2 3 ~3 4 t2 t 3 t t 2 t2 3 t2 2

If the canonical commutation relations (1.8) hold at latticesite 1, one can verify by direct calculation using (4.6) and(4.7) that relations (1.8) hold at lattice site 2, through or-der h . We remark that, because the lattice equationsagree with the continuum equations through order h, un-itarity is assured through order h . That is, the calcula-tion described above verifies that Eqs. (4.5) preserve uni-tarity through the first nontrivial order in the lattice spac-ing. If we had not symmetrized H as in (4.2), unitaritywould fail at order h .

It is important to emphasize that once we have shownthat the canonical nature of the variables c,c,d, d ispreserved as one advances from lattice site I to lattice site2, it follows immediately that the SU(2) commutation re-lations of the spin variables are preserved as one goesfrom lattice site 1 to lattice site 2.

V. PROPERTIES OF THE SUBSTITUTION OPERATOR

In view of the discussion in Sec. IV, it is clear that theproblem of showing that the unitarity of a discretized spin

system is preserved reduces to a study of Hamiltoniansthat are totally symmetric in pairs of canonical variables.It is sufficient and notationally more convenient to con-sider Hamiltonians of the form

H =H(q, p;r, k) . (5.1)

Here instead of using the creation and annihilation opera-tors we use the Hermitian, canonically conjugate pairs ofvariables (q,p) and ( r, k), where

c+c d+dq =-

2'

2) 7

(5.2)p=, k=C —C

i2 ' i2The canonical commutation relations are

[q,p] = [r,k] =i,[q r]=[p k]=[q k]=[rpl=o.

(5.3)

To analyze Hamiltonians of the type (5.1), it is helpfulto develop the mathematical properties of the substitution


[q p ]='D, (p (5.4)

Using the operator power-series representation for g (p)one finds

[q,g (p)] =iD„,g (p) =iD» ~

From this one can establish, by induction, that

(5.5)

operator, Dq„ introduced and defined in Sec. III. Weremind the reader of the elementary properties given in(3.3). As we saw in Sec. III, the substitution operator D,is a generalized derivative operator. Since it acts on poly-nomial functions of operators, it provides a convenientway to represent certain commutators and it satisfies aninteresting array of identities, some of which are ratherelaborate. Some of these identities are generally truewhereas others only hold under restricted conditions.

One can prove a set of identities for the substitutionoperator D if the arguments of the operator polynomialsare canonical variables such as those in (5.3). It is wellknown that [q,p "]=inp" ' which, in the D notation is

where I }»z is the Poisson brackets and

D,,;f=[qf]-[qf},,,=(~ a

D,,;f= —[p,fl —[p f }»,p =&qp pq

The results above generalize in an obvious way when f, gdepend upon more than one pair of canonical variables.

When the commutator of the operators is not a c num-ber, say [s,t] =a, it is not difficult to show that

[s,f(s, t)]=D, ,f(s, t) .

In fact, one can obtain a more general result. Consider aset of n operators (r(, r2, . . . , r„) which need not com-mute and need not be distinct. Let s be another operator,possibly a member of the set, and let [s,r; ]=b;. Obvious-ly [s,r;]=D„b r;, and

[~ "i"j 1=g Dr„,t „"i"j

k

[q",g'(p)] = D» r( g(p(q'" .

It follows that

(5.6)By induction it follows that

[f(q»g (p»)]=iD, , D„g( X(q)

=iDp c) f(»g(p) ~ (5.7)

where the second form can be obtained by interchanging fand g. The same techniques can be used to obtain a moregeneral result. Given an operator function f(q,p), onecan use [q,p]=i to write its power series asf ( q,p) =g f„q"p . From (5.7) one gets

ff (q p),g (q p)] = ~ g f. gkI [ q "(D ~ e')p™

k=1

This provides a general representation for the commutatorof s with a polynomial function of any number of opera-tors.

We list, without proof, some useful identities in thegeneral case when the commutator of the operator vari-ables is not a c number, and indicate under which cir-cumstances they hold.

First we mention an identity that holds without any re-strictions:

n(D k) I]7 D, ,D, b

—D, bD, =Dt g b—D, D (5.12)

Because of the linearity properties (3.3) this can be writtenin the form

[f(q*p'»g (q p»)l =& (D,n, ,fg D,D fg)—=i (D» g) (sf DI, p (sf)— (5.8)

where the second form is obtained by interchanging f and

The identity (5.12) is the generalization of the commuta-tive property of partial derivatives. Under the correspon-dence (5.10) between the operator D, , and partial deriva-tives, it becomes

a„a „a a aha „aa a

Dp, D &H~ Dq, D &H INotice that under the correspondence rule

(5.9)

If one sets f=g =H in (5.8), the trivial statement[H,H] =0 gives

In (5.12) the D operators are assumed to be applied tofunctions of the noncommuting variables s and t, while aand b are operators that can be functions of s and t andneed not commute with s, t or with each other. A specialcase of the identity (5.12) is obtained when a = 1:

aDs a~a

Bs(5.10)

s, 1Dt, b Dt, b s 1+Dt, D lb

This relation is the generalization of

(5.13)

the right-hand side of (5.8) reduces to

iaf ag af iagaq ap a~ aq

That is, under the correspondence rule (5.10),

[f(q p'»g(q p)l I»g }»,p (5.1 1)

c) (3 06as at

=at as Os a~

+

in calculus.Next, we give an example of an identity that is valid

only when acting on a polynomial that is a totally sym-metric function of its operator variables:


Ds, aDt, 1 t, aD$, 1(5.14)

We illustrate (5.14). Consider the function H =st Onone hand, we have D, ,D, 1H =2at while, on the otherhand, D, ,D, 1H =at+ta. However, if H is totally sym-metrized: H =st +tst+t s, then D, ,D, 1H =D, ,D, 1H= 3( ta +at).

More generally, let S be the symmetrizing operator,

[qi,pi] = [r, ,k, ]=l,[qi ri]=[pi, kil=[qi, kil=[s» ri]=O,

(6.1)

(6.2a)

they continue to hold exactly at lattice site 2. The latticedifference equations of motion that determine operators atsite 2 in terms of operators at site 1 are

P2 —P1 H(—q,p;r, k), (6.2b)where P is a permutation operator and gp extends overthe N1 possible permutations of N objects. ' Then the to-tally symmetric polynomial of degree n in t and degree min s is St"s . Clearly, D, ,D, &(St"s )=mnSat" 's

is the same as D, ,D, i(St"s ) In th. e above, we rely uponthe fact that the operator D, , commutes with S. This re-sults from the linear nature of the operators D and S.

It is interesting to note that the central result of Ref. 10is an immediate consequence of the identity (5.14). Con-sider a Hamiltonian which is a function of a single canon-ically conjugate pair of variables (q,p). The continuumequations of motion are

r& —r1

h=Hk(q, p;r, k), (6.2c)

kq —klh

H„(q,p—;r,k) . (6.2d)

H =D ]H

[compare (5.16)], and we use

(6.3a)

As in Sec. III, P=(pi+p2)/2, etc. , and we have abbrevi-ated

q=Dp1H, p= —Dq 1H . (5.15) H=H(q, p;r, k) . (6.3b)

The lattice transcription of (5.15) is

q2 —q1 =D,H(q, p), (5.16a)

Following the procedure used in (5.17), we commute (6.2a)on the right with P, and (6.2b) on the left with q, and addthe resulting equations:

p2 pi-q, ] (5.16b)

To prove unitarity, we commute (5.16a) with P on theright and we commute (5.16b) with q on the left. Addingthe results gives

h[q2 pz] —[qi pi]= —([H- p] —[q,H ]) . - (6.4)

This procedure can be followed with all pairs of variables,and therefore the requirement of unitarity is equivalent tothe following six equations:

[H,p] = [q,H ], - (6.Sa)

2([q2 pal —[qi pi])= D- D- iH(q, P)

D ,D,H(q—P )-,

where

(5.17)

[Hq, k]=[r,H ],[r,H ]=[q,Hq],

[H,k]=[q,H„],[Hk p]=[r,H;],

(6.5b)

(6.5c)

(6.5d)

(6.5e)a =[qP]. (5.18)

[q2 p21=[q»pi] . (5.19)

Note that a is an operator and not a c number, because qand p are not canonically conjugate. We now observe thatif H is a totally symmetric function of q and p, we canuse (5.14) to conclude that the right side of (5.17) van-ishes, thus establishing that

[H, k]=[H„,p] .

For example, when H =q pk Eqs. (6.2) can be solved ex-actly, and the consistency conditions (6.5) can be verifiedexplicitly.

Using the equations of motion (6.2) and the identity(5.8) we can rewrite (6.5a) in the substitution operator no-tation of Sec. V as

This immediate proof of unitarity demonstrates the powerof the substitution operator calculus.

Dq )q+Dp tp=2(D D p Dn —p-—+Dk, D„„-p—D,,D„„-p» (6.6)

VI. UNITARITY FOR 0 ( q, p; r, k ) ON THE LATTICE

As we have seen, proving unitarity for a general spinsystem on the finite-element lattice is equivalent, via theSchwinger representation (1.7), to proving lattice unitarityfor a general Hamiltonian of two pairs of canonical vari-ables (q,p) and (r, k) Our objective is. to show that if thecanonical commutation relations hold at lattice site 1,

0q dp Bq Bp+Bq ap

'aqap ap aq+ ar ak

Bq Bp(3k Br

(6.7)

where q =q1, p =pl, etc.It is illuminating to rewrite (6.6) for the case of com-

rnuting variables:


Note that Eq. (6.2a) can be rewritten as

h-q=q+ —H (6.8)

Applying the four substitution operatorsD ~, D &,D~ &, Dk

&to (6.8) yields the system of equations

1 Dq ]q + Hh—

O=D„)q +—Hh—

O=D )q+ —Hh—

h-O=Dk )q + —H

(6.9)

Once again, it is illuminating to write the equations forcommuting variables:

Bq h BHBq 2 BpBq

Bq h BH+Br 2 BpBF

Bq

Bp

Bq

Bk

h dH+(3p

h BH+—2 apeak

(6.10)

h BH1 ——

2 Bpbq

h $2H2

Bq

h BH2 Bkhq

h BHBrBq

h BH2 Qp

h BH1+—2 Bqdp

h

2 akap

h 02H

2 Brhp

h

2

h1 ——

2

c)p ()r

(3 Hdqdr

BHaSar

h dH2 Bv

h BH2 BpBk

h BH2 dqBk

h BH2 Bk

h BH1+—2 Qr3k

(6.11)

It is clear now what a proof of unitarity entails. Onemust invert Eqs. (6.9) plus the twelve other equations ofthe same type to solve for Dq &q, etc. , and to substitute thesolutions into (6.6) to demonstrate the validity of the uni-tarity consistency condition. To illustrate this process, letus continue with the commuting forms of these equations,(6.7) and (6.10).

In this case we merely invert the matrix:

spin variables. This problem is an important step in ourprogram, which is aimed at obtaining a consistent latticeformulation of quantum field theories without ferrniondoubling and which is amenable to direct numerical solu-tion without employing Monte Carlo techniques. Thefinite-element method produces a discrete quantum theorythat is fully consistent for finite lattice spacing. Includinginternal degrees of freedom, such as spin, is highly non-trivial.

Surprisingly, we find that no direct discretization of thecontinuum equations of motion for the spin variablesobeying the algebra of SU(2) can lead to a fully consistentunitary quantum theory. To formulate such a discretetheory we are required to introduce a more fundamentalset of variables. We find it natural to choose theSchwinger representation of the spin variables in terms ofboson creation and annihilation operators. Once theHamiltonian is rewritten in terms of these variables itmust be fully symmetrized to produce lattice equations ofmotion that preserve unitarity. These lattice operatorequations of motion for the oscillator variables are ine-quivalent to the lattice equations of motion for the spinvariables provided directly by the finite-element prescrip-tion. This choice is attractive because there is a well-known generalization to the algebra of SU(N) that em-ploys N(N —1) one-dimensional oscillators or N —1 os-cillators ig X dimensions. ' Thus we expect the resultsobtained here to generalize in a straightforward way totheories with internal degrees of freedom described bySU(N).

ACKNOWLEDGMENTS

We thank J. Louck for helpful conversations. C.M.B.,K.A.M. , S.S.P., and L.M.S are grateful to the AspenCenter for Physics, where part of this work was done.C.M.B., K.A.M. , and S.S.P. also acknowledge the hospi-tality of Los Alamos National Laboratory. This workwas supported by the U.S. Department of Energy.

Dp~ Hp ——Dq0 Hq .PP

(6.12)

We have established the consistency condition in severalexamples, but have not yet completed a proof to all ordersin h.

A straightforward but tedious calculation shows thatindeed the consistency condition (6.7) is satisfied.

This procedure is much more difficult to implement foroperators. The most promising scheme is based on ex-panding the operators in powers of the lattice spacing h,as was done in Sec. IV. Properties of the substitutionoperator discussed in Sec. V must be used extensively, inparticular ones that depend on the total symmetry of H,such as (5.14) and

APPENDIX: SPIN ALGEBRA POLYNOMIALS

In proving unitarity of the spin Hami1tonians, sym-metric combinations of the oscillator variables played acrucial role. There are two pairs of creation and annihila-tion operators which are relevant, c, c, d, and d . Weare led, as in Ref. 10, to seek relations between symmetricpolynomials in these variables.

The procedure by which the polynomials in questionare generated is to rewrite symmetric polynomials in c, c,d, and d as polynomials of a fundamental quadralinearpolynomial in these four variables. This basic polynomialis constructed from

VII. CONCLUSIONS

In this paper we have studied the finite-element methodas applied to Hamiltonians that are arbitrary functions of

T» ——c c +cc and U& &

——d d +dd

by multiplication:

X'= T)( U)( =4ab +4ba +1=1+4(s—z ),where we have used the construction (3.19), with

(Al)

(A2)


a = (S„ i—S~)= d c,1 . 1

b= (S +iS~)= c d,1 . 1

v'2 2

and where s is the Casimir operator,

s:S~ +Sy +Sz

and z =S,. The biquadratic polynomials

(A3)

(A4)

Po ——1,Pi ——X,P2 = (X + 16s)—2X +5,P, = (X'+ 80sX) —10X'+45X,

P4= (X +224sX +2304s )+(—28X —576sX)

+ ( 270X + 3 168s ) —396X +729 .

By rescaling X,

X =4~sy,and writing

(A9)

(A10)

T2p = C C CC +CCC C +CC 'CC +C CC C

+cc c c+c ccc =6S(c c cc),U22—:6S(dtd dd),

where S is the symmetrization operator, lead to

T22 U22 = —', (X —2X +. 16s + 5 ) .

Similarly, from

(A5)

P„(y)=4" g s'" "1 W„k(y),k=0

(A 1 1)

Wo, o(y) =1

W1,o(y) =y

W2, 0(y) =y'+, W2, 1(y) W2, 2(y)—

we generate the spin polynomials W„k(y), the first few ofwhich are

T33 ——20S(c c c ccc),

U33=20S(d d d ddd),

we construct

T33 U33 —,(X —10X +45 +80sX) (A8)

W3 0(y) =y'+ 5y', W3 1(y) = ——', y',W3 2(y) = —,„y, W3 3(y) =0,W40(y) =y"+14y'+9 W41(y) = —7y' —9y,

135 P 99 99W4 2(y) =, y + —, , W4 3(y) = ——„y

7294(y) =

(A12)

By looking at the explicit decomposition of T„„U„„interms of X we generate the polynomials P„. If we nor-malize these polynomials so that the coefficient of X" isone, then, explicitly, the first few are

It will be noticed that W„o are the continuous Hahn poly-nomials, ' as must arise, since in the s~ oo limit the spinalgebra reduces to the Heisenberg algebra. It is nontrivialto derive the recursion relation which allows us to gen-erate all the spin polynomials:

W„k =yW„1k+(n —1) W„2 k+ „(n —1) (2n——6n +9)W„

, (n —1) y W„2—k 1+ „(n —1) (n ——2) y W„3(n 1 ) (n 2) (n 3) W& 4 g 4& n &0

The generating function for the W„k's is the polynomial P„:

(A13)

P„(t,y) = g t"W„k(y),k=O

which satisfies the recursion relation

(A14)

2n —6n +9 ~ ytP„=yP„1+(n —1) 1+16

t ——P„2

(n —1) (n —2) 2 (n —1) (n —2) (n —3)+16 y "-' 256

'n 4~tP — tP n)0. (A15)

We finally remark that none of the sets of polynomials [ W„k, k =O, n I, [ W„k, n =0, co I is orthogonal except for theHahn polynomials [ W„o,n =0, oo I.


'C. M. Bender and D. H. Sharp, Phys. Rev. Lett. 50, 1535(1983).

C. M. Bender, K. A. Milton, and D. H. Sharp, Phys. Rev. Lett.51, 1815 (1983).

C. M. Bender, K. A. Milton, and D. H. Sharp, Phys. Rev. D31, 383 (1985).

4C. M. Bender, K. A. Milton, D. H. Sharp, L. M. Simmons, Jr. ,and R. Stong, Phys. Rev. D 32, 1476 (1985). We refer to thispaper as I.

5C. M. Bender, F. Cooper, V. P. Gutschick, and M. M. Nieto,Phys. Rev. D 32, 1486 (1985).

F. Cooper, K. A. Milton, and L. M. Simmons, Jr. , Phys. Rev.D 32, 2056 (1985).

7C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons,Jr., Phys. Rev. D 33, 1692 (1986). We refer to this paper asII.

C. M. Bender, L. M. Simmons, Jr., and R. Stong, Phys. Rev. D33, 2362 (1986).

C. M. Bender and M. L. Green, Phys. Rev. D 34, 3255 (1986).C. M. Bender, L. R. Mead, and S. S. Pinsky, Phys. Rev. Lett.56, 2445 (1986).C. M. Bender and K. A. Milton, Phys. Rev. D 34, 3149 (1986).K. G. Wilson, Phys. Rev. D 10, 2445 (1974).

t J. Schwinger, in Quantum Theory of Angular Momentum,edited by L. C. Biedenharn and H. Van Dam (Academic, New

York, 1965), p. 229."J. D. Louck, J. Math. Phys. 6, 1786 (1965); J. D. Louck and

H. W. Galbraith, Rev. Mod. Phys. 44, 540 (1972).'5The properties of the operators P and 5 are discussed, for ex-

ample, in A. Messiah, Quantum Mechanics lNorth-Holland,Amsterdam, 1962), Vol. II, Chap. 14.

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Discrete-time quantum mechanics. III. Spin systems