Quantum Electrodynamics on a Space-timeLattice

Jerzy Kijowski and Artur ThielmannCenter for Theoretical Physics, Polish Academy of Sciences

Al. Lotnikow 32/46; PL-02-668 Warsaw, Poland

Abstract

A Minkowski-lattice version of quantum electrodynamics (or rather its simplifiedversion, with matter described by a scalar field) is constructed. Quantum fieldsare consequently described in a gauge-independent way, i.e. the algebra of quantumobservables of the theory is generated by gauge-invariant operators assigned to zero-,one-, and two-dimensional elements of the lattice. The operators satisfy canonicalcommutation relations. The uniqueness of representation of this algebra is proved.Field dynamics is formulated in terms of difference equations imposed on the fieldoperators. It is obtained from a discrete version of the path-integral. The theory islocal and causal.

1 Introduction

Standard difficulties of quantum field theory (renormalization, regularization of the prod-uct of fields at the same space-time point etc.) are augmented in the case of gauge fieldsby the gauge-fixing problem. Having at our disposal no satisfactory non-perturbativedescription, one commonly uses non-physical objects like ghosts, bare particles etc.

In the present paper we show how to solve completely the gauge-invariance problem,at least at the level of lattice approximation, for the theory of interacting scalar chargedfield φ and the electromagnetic field, described by the vector-potential Aµ. We use thereal-time lattice, not its euclidean version. This way we maintain the essential structureof the Quantum Field Theory: the local character of quantum field operators fulfilling thecanonical commutation relations, local time evolution given by a unitary transformation,the causal (light-cone) character of field dynamics. As a result we obtain a discreteversion of an algebraic Quantum Field Theory, where, similarly as in Haag formulation(see [7]), the quantum field is described in terms of algebras of local observables, assignedto bounded regions of the lattice. The quantum dynamics is formulated in terms ofdifference equations, which have to be satisfied by the field operators. The equations

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have causal character and the initial value problem is well posed. In particular, operatorsdescribing electric and magnetic field fulfill the Maxwell equations with the continuousoperators div and curl replaced by their obvious lattice analogs.

Of course, the problem “how to remove the cut-off introduced by the finite latticespacing and to construct an exact, continuous model of interacting gauge-fields?” isseveral orders of magnitude more difficult. We do not discuss it here. But we believethat, due to its simple structure, our model may be very useful for understanding thefundamental structures of the quantum gauge field theory.

Our construction is based on the following, heuristic point of view, which we foundextremely useful as a guiding principle. Consider two different Cauchy surfaces: Σinit ={x0 = tinit} and Σfin = {x0 = tfin} in the Minkowski space-time M . The classicaldynamics gives the relation between the initial data on Σinit and the final data on Σfin.We want to define its quantum counterpart by the path integral

(Ψfin|Ψinit) =1N

∫Ψfine

ih

∫Ld4yΨinit

∏xfin

dA(xfin)dφ(xfin)dφ(xfin)×

× ∏y

dA(y)dφ(y)dφ(y)∏xinit

dA(xinit)dφ(xinit)dφ(xinit) . (1.1)

Here, the initial (resp. the final) quantum state Ψ is represented by a “wave function” Ψdepending on initial configurations A(xinit) and φ(xinit) (resp. final configurations A(xfin)and φ(xfin)). The functional integral may be thought naively as being performed overthree groups of variables: initial configurations (A(xinit), φ(xinit)), intermediate configu-rations (A(y), φ(y)) and the final configurations (A(xfin), φ(xfin)). Performing the inte-gration over the first and the second group, with the last one being fixed, should give usthe unitary evolution kernel U(Σfin,Σinit) from Σinit to Σfin:

U(Σfin,Σinit)Ψinit =1N

∫eih

∫Ld4y Ψinit

∏y

dA(y)dφ(y)dφ(y)×

× ∏xinit

dA(xinit)dφ(xinit)dφ(xinit) . (1.2)

The goal of the present paper is to construct a rigorous, discrete version of the above func-tional integral. For this purpose let us imagine that the region of space-time containedbetween Σinit and Σfin is divided into small four-dimensional hypercubes. This discretiza-tion induces also a partition of both Σinit and Σfin into three-dimensional cubes. In adiscretized version of the theory the quantum state on any Σ will be defined as a wavefunction depending on a discrete number of degrees of freedom related to those three-dimensional cubes. We define the physical quantum states as those, which fulfill thediscrete version of constraint equations, relating the divergence of the electric inductionfield to the electric charge carried by the matter field (see [4] and the discussion therein).At the level of the continuous theory this constraint is described by equation (2.11) andby equation (4.17) in the discrete version. This requirement implies, that the physically

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admissible wave functions have to be gauge-invariant on Σ’s. Hence, they can be consid-ered as functions of a complete set of gauge-invariants. The first step of our constructionconsists in organizing such gauge-invariant functions into a Hilbert space. This way weobtain a lattice version of gauge-invariant quantum kinematics.

To define the dynamics of such a theory we have to pass through the following threesteps:

1) We discretize the action on the lattice. Since the action does not change undergauge transformations, it can also be expressed in terms of a complete set of gauge-invariant observables.

2) We have to remove from (1.2) an infinite factor, resulting from the integration over thegauge parameters, and to leave only the integration over “true degrees of freedom”.This has to be done not only for intermediate configurations (A(y), φ(y)) but alsofor the initial configurations. This way the initial arguments of the evolution kerneldefined by (1.2) match the (gauge-invariant) arguments of the initial wave-functionrepresenting the quantum state of the system.

3) Since both the initial wave function and the Lagrangian are gauge-invariant, we obtainfrom (1.2) a gauge invariant final wave function representing the physical quantumstate.

The integral we use in the second step of this procedure is, in fact, defined on the“gauge-orbit space”, i. e. over the quotient of the configuration space modulo the actionof the gauge group. There have been many proposals to define a natural measure on thisspace. In the present paper we use the measure proposed in [19]. But we prove in Section4.3 a Theorem about the uniqueness (up to equivalence) of the irreducible representationof the observable algebra obtained this way. This result fully justifies, in our opinion, thecorrectness of our approach.

It turns out that the dynamics defined this way splits in a natural way into a superpo-sition of the “kinetic” evolution and the “potential” evolution. When the time-spacing τtends to zero, the Trotter formula gives us the (spatially-discretized) Hamiltonian, whichis a sum of the kinetic and the potential energy of the field.

Mathematically, our construction is based on a lattice approximation for gauge fields(see [8] and [10]) where, like in the continuum theory (and unlike in Wilson’s approach,cf. [18]), the gauge potential A is described by Lie-algebra (not Lie-group) elements as-signed to the one-dimensional elements (“links”) of the lattice. As a consequence, alsothe electric induction field Dk becomes a self-adjoint operator with continuous spectrum– like in the continuous version of the theory. However, the “longitudinal part” of theelectric induction field, corresponding to electric charges, should have a discrete (“quan-tized”) spectrum. In our construction this phenomenon is due to the gauge invariance ofthe physical wave functions. As will be seen in Section 4, the gauge invariance implies thecompactification of some degrees of freedom of the field. They are no longer described by

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an R1-variable but by a U(1)-variable. This non-trivial topology of the reduced configu-ration space is responsible for the quantization of the corresponding conjugate momenta,which turn out to be exactly electric charges, carried by the matter field. Reducing con-sequently both the Hilbert space of quantum states and the Feynman kernel with respectto the gauge group, we end up with the gauge-invariant version of the theory, describedessentially in terms of the quantities arising in its continuous version (cf. results obtainedin [14] and [15] for the continuous, spinorial electrodynamics).

The paper is organized as follows. In Section 1 we define the model. In Section 2we present our lattice approximation and reduce it with respect to the gauge group. InSection 3 we show how the classical dynamics of the model may be completely described interms of a complete system of gauge invariant observables. Also, we prove the causality ofthe dynamics. In Section 4 we construct the Hilbert space of (gauge-invariant) quantumstates of the field i.e. we find the representation of the reduced canonical commutationrelations. In Section 5 we derive the quantum dynamics of the model from the gauge-invariant version of the Feynman integral, with few more technical calculations postponedto section 6. Finally, in Section 7 we give an equivalent formulation of the theory, in termsof the configuration variables only, without any use of the canonical momenta.

We consider the present result as an introductory step towards the construction ofthe exact spinor electrodynamics on a lattice, which we are going to present in the nextpaper.

2 Continuous version of the theory

2.1 Definition of the model

We consider the theory of a complex-valued matter field φ interacting with the electro-magnetic field represented by the potential Aµ. The theory is defined by the Lagrangian

L = −V(|φ|2)− 12DµφD

µφ− 14fµνf

µν , (2.1)

where

Dµφ = ∂µφ+ ie

hAµφ

fµν = ∂µAν − ∂νAµ ,(2.2)

and V is a function of a real variable (e.g. a Higgs potential). To simplify the formulaewe will denote by g = e

hthe electromagnetic coupling constant. The metric tensor is

equal gµν = diag(−1, 1, 1, 1). The gauge group U(1) acts on the space of field configura-tions as follows:

φ = e−iλφ

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Aµ = Aµ +1g∂µλ

(2.3)

Variation of the Lagrangian (2.1) with respect to the matter field leads to the second-order, non-linear Klein-Gordon equation

DµDµφ− 2 V′ φ = 0 (2.4)

for the matter field, whereas variation with respect to the gauge potential gives theMaxwell equation (we use Gauss system of units).

∂νfµν = −g Im

(φ Dµφ

), (2.5)

with the right-hand side being the electric current jµ carried by the matter field φ.

2.2 Gauge freedom reduction

Configurations which differ only by a gauge transformation are physically equivalent.To describe the “true degrees of freedom” of the field we have to pass to the quotientspace with respect to this equivalence. This may be done e.g. by a gauge-fixing. For ourpurposes we use, however, a different description, namely a description in terms of the socalled hydrodynamical invariants (see [3], [11], [8])

R = |φ| ,vµ =

1ig

φ

|φ|Dµφ

|φ| = Aµ +1g∂µ(argφ) .

(2.6)

Inserting these invariants to equations (2.4) and (2.5) leads to the following field equations

∂µ∂µR− 2V′ R = g2vµv

µR ,

∂νfµν = −g2R2vµ .

(2.7)

The name “hydrodynamical invariants” is justified by the fact that (2.7) may be inter-preted as the equation of motion for a charged fluid of density R2 and velocity vµ, having anon-standard constitutive equation which is implied uniquely by the function V (see [3]).

For field configurations with non-vanishing φ, the values of the hydrodynamical in-variants enable us to reconstruct the field configuration (φ,Aµ) uniquely up to a gaugetransformation and the field equations (2.7) are equivalent to the original equations. Fora generic configuration, vanishing of φ along a two-dimensional world sheet implies thatthe field vµ carries a vortex, i.e. the singularity of the curl of v see [3] and [11]. The vortex

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has to fulfill the Dirac’s quantization condition (see [6]). Under this condition, the re-construction (up to a gauge-transformation) of the field configuration from the invariantsis, again, possible. There is a conservation law for the vortices. Hence, describing fielddynamics, we have to take into account the “string-like” degrees of freedom carried bythe vortices. This is relatively difficult in the continuous version of the theory, but maybe done in a completely satisfactory way in its discretized version.

2.3 Phase space of initial data

Given a Cauchy-hypersurface {x0 = const}, the initial data for the theory are describedby the values of the fields φ,Aµ and their conjugate momenta

π0 = 2∂L

∂{∂0φ}= −D0φ

p0µ =∂L

∂{∂0Aµ} = −f 0µ = f0µ

(2.8)

on the hypersurface. The factor 2 in the definition of π0 appears, because

π0 = Re π0 + iIm π0 , (2.9)

where

Re π0 =∂L

∂(∂0Re φ)(2.10)

is the momentum conjugate to Re φ and analogously for the imaginary part.Vanishing of the momentum p00 (Dirac’s primary constraint) implies the reduction

of the phase space with respect to A0. This way we obtain the reduced phase spacedescribed by the quantities (φ,Ak, π0, p0k), k = 1, 2, 3. There remain, however, secondaryconstraints, implied by the zero component of the field equation (2.5):

1g∂kp

0k − Im(π0φ

)= 0 . (2.11)

It is easy to see that the above constraint is equivalent to the statement that initial data,which differ only by a gauge transformation on the Cauchy hyperplane are equivalent. Thenaive proof of this statement consists in reparameterizing the initial configurations (φ,Ak)by the invariants (R, vk) and the phase of the matter field α := argφ. This is a pointtransformation in the (infinite dimensional) symplectic space of Cauchy data. The corre-sponding transformation for the momenta can be obtained by rewriting the pre-symplectic

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form in terms of new variables (cf. [5], [9]). Using (2.6) and integrating by parts we obtainthis way:

(Re π0)δ(Re φ) + Im (π0)δ(Im φ) + p0kδAk =

=12

(π0δφ+ π0δφ) + p0kδAk =

=(

1g∂kp

0k − Im (π0φ))δα + Re (π0φ)

δR

R+ p0kδvk − 1

g∂k(p0kδα) .

(2.12)

The last (boundary) term vanishes when integrated over the entire Cauchy hyperplane.Hence, the gauge-invariants (p0k,Re (π0φ)) are the momenta canonically conjugate to theinvariants (vk, logR), whereas the vanishing left-hand side of (2.11) becomes the momen-tum canonically conjugate to the (completely arbitrary) phase of the matter field φ. Thesecondary reduction yields therefore the phase space described by the gauge-invariant,unconstrained observables (p0k,Re (π0φ), vk, logR). Knowing them, it is again possible toreconstruct uniquely (up to a gauge transformation) the entire non-reduced initial data(p0k, π0, Ak, φ).

The above, naive picture may become a rigorous result if we also describe the topo-logical degrees of freedom carried by vortices corresponding to zeros of the matter field φ(see [11] for the case of electrodynamics and [12], [13] for the case of a non-abelian gaugegroup). In the present paper we are going to construct the lattice version of this rigorousresult.

3 Lattice approximation

3.1 Structure of the lattice

We construct the approximation of this theory on a four-dimensional lattice Λ in Min-kowski space. We recall, that the metric tensor is diag(−1, 1, 1, 1). As Λ we take theCartesian product of the three-dimensional, cubic lattice Σ and the discrete time axis T .Thus, the lattice sites are the points x = (τn0, an1, an2, an3) ∈ R4, where τ denotes thelattice spacing in the time direction, a denotes the lattice spacing in the space directionsand n = (n0, n1, n2, n3) ∈ Z4 is a point with integer coordinates. Sites of Λ will be denotedby x, . . ., links (x, x + µ), . . ., plaquettes (x; µ, ν), . . . , where µ is a vector of lengtha(µ) (i.e. τ for µ = 0 and a for the space dimensions) and direction of the oriented µ-thaxis. We adopt the following conventions for summation ranges

µ, ν = 0, 1, 2, 3µ, ν = 0,−0, 1,−1, 2,−2, 3,−3

(3.1)

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k, l = 1, 2, 3k, l = 1,−1, 2,−2, 3,−3 .

(3.2)

We will begin with a spatially-bounded lattice Σ. The evolution of the field over sucha lattice is not uniquely determined by the evolution equations unless we fix boundaryconditions for the field in an appropriate way. Here, we do not impose any non-physical(e.g. cyclic) boundary conditions but we only fix the value of the field on the externalsites and links of Σ.

There is a natural, inductive relation between the theories obtained for Σ1 and Σ2,when Σ1 ⊂ Σ2. Using this relation, as a last step of the construction of the field the-ory, we may pass to the inductive (thermodynamic) limit and construct this way thecorresponding theory for the complete (unbounded) lattice Z4.

3.2 Field configurations on the lattice

We will represent the fields (φ,Aµ) on the lattice in the following way. The matterfield will be represented by its complex value φx at every site x. The gauge field willbe represented by the mean value Ax,x+µ ∈ u(1) = R1 of the component Aµ of theelectromagnetic potential, on the link (x, x+ µ).

The gauge group of the theory is described by real-valued functions of lattice sites:λx ∈ u(1) = R1. A gauge transformation is defined as a discrete version of (2.3):

φx = e−iλxφx ,

Ax,x+µ = Ax,x+µ +λx+µ − λxga(µ)

,

(3.3)

The description of the gauge potentials and gauge transformations in terms of elementsof the Lie-algebra u(1)= R1 and not the Lie-group U(1), has been proposed in [10].Heuristically, such a description was motivated by the geometric structure of the theory:for each lattice link (x, x + µ) it is possible to choose an internal gauge transformation(i.e. a gauge which is trivial on the link’s ends) which makes the corresponding Aµ constanton the link. In the lattice approximation of the theory we want to keep this property.Hence, we describe the gauge field by this constant value Aµ, instead of replacing it byits truncated value eiga(µ)Aµ .

To remain consistent with the above picture, gauge transformations on the lattice haveto be considered as restrictions of continuous gauge transformations to the lattice sites.Therefore, it does make a difference whether we gauge the field at the point x by 0 or

8

by 2π, because, after interpolation of the transformation to the entire link (x, x+ µ), theresulting value of Ax,x+µ will be transformed in a different way.

We define the following lattice expressions for the covariant derivative Dµφ of thematter field and for the curl fµν of the gauge field:

(Dφ)x,x+µ =1

a(µ)

(eiga(µ)Ax,x+µ φx+µ − φx

),

fx;µ,ν =1

a(µ)a(ν)

(a(µ)Ax,x+µ + a(ν)Ax+µ,x+µ+ν +

+ a(µ)Ax+µ+ν,x+ν + a(ν)Ax+ν,x

).

(3.4)

It follows immediately that f , being the lattice curl of A, is invariant with respect to gaugetransformations (3.3) and that (Dφ)x,x+µ behaves like φx under these transformations.

3.3 Reduction of gauge freedom

Physical field configurations are described by the space of gauge orbits, i.e. the quotientspace of the above field configurations modulo gauge transformations. We will use dif-ferent, equivalent parameterizations of this quotient space. Below we give a list of foursuch parameterizations. In fact, we need only the first and the last of them. The firstparameterization is closely related to the gauge-dependent parameters (φ,A). The lastone gives the final set of parameters, which we use to describe the theory. The easiestway to pass from one to the other (e.g. in the functional integral) is to use the remainingtwo parameterizations as intermediate steps.

Parameterization 1. (“tree-gauge”) If there is a gauge-condition such that eachclass of gauge-equivalent configurations contains one and only one representative fulfillingthis condition, the entire class can be parameterized by this particular representative. Aconvenient way to fix a gauge consists in fixing the values Ax,x+µ of the gauge field for allthe links (x, x+ µ) belonging to a set T of links which has properties of a tree (see [12]).By a tree we mean a pair (x0, T ), where x0 ∈ Λ is a lattice site which we call a root ofthe tree and T is a subset of links of the lattice, having the following property: for anysite x ∈ Λ there is one and only one path connecting x with the root x0 and composed oflinks belonging to the tree. We denote this path by (x0, x)T .

By a tree-gauge we mean fixing the value of all the gauge potentials along the tree(e.g. putting them equal to zero). Now we may use all the remaining parameters (i.e. thevalues of the matter field φx and the values of the gauge potential Ax,x+µ on all the linkswhich do not belong to the tree) as parameters in the space of gauge orbits. There is,however, a residual gauge which still remains: the global gauge transformation whichrotates all the values φx in the same way and does not change the values of the gauge

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potential. To remove this freedom we may fix the phase of the matter field at the root(assuming e.g. that φx0 is real).

A most simple example of a tree can be obtained as follows. Choose any root and takeas T the following collection of links

1) all the links (x, x+ 3) belonging to the x3-axis passing through the root,

2) all the links (x, x+2), belonging to the two-dimensional plane (x2, x3) passing throughthe root,

3) all the links (x, x + 1) belonging to the three-dimensional plane (x1, x2, x3) passingthrough the root and, finally,

4) all the time-like links (x, x+ 0).

Fixing A0 is usually called a temporal gauge. This is however not a complete gaugecondition, since any gauge transformation, which is time-independent leaves the conditionunchanged. Fixing also gauge fields on the remaining space-like links of the above treeand fixing the phase of the matter field at the root, removes this residual gauge-freedom.

A three-dimensional tree defined by 1) – 3) can also be used to fix completely a gaugein the space of Cauchy data on a fixed three-dimensional lattice-layer {x0 = const}. Thisis a generalization of an axial gauge-condition.

Parameterization 2. (“improved tree gauge”) Keeping a tree gauge, we may pa-rameterize the electromagnetic degrees of freedom by the values of the field strength f .However, not all the values of fx;µ,ν are independent, because f is a closed form. Thismeans, that the sum of f over all the plaquettes belonging to the boundary of any three-dimensional cube vanishes. To find independent parameters we take for every off-tree link(x, x+ µ) the closed path composed of 1) the tree-path (x0, x)T 2) the link (x, x+ µ) itself3) the inverse tree-path (x+ µ, x0)T . The above closed path is the boundary of a surfacecomposed of a finite number of “along-tree-plaquettes”. Take the value Fx,x+µ(T ) of theelectromagnetic flux through it, i.e. the sum of all the values fx;µ,ν corresponding to thissurface. The collection of fluxes Fx,x+µ(T ) corresponding to all the off-tree links containsthe complete, independent information about the gauge field. To prove this statementwe observe that Fx,x+µ(T ) is equal to the value of a(µ)Ax,x+µ plus the sum of all thealong-tree values of the gauge potential A (multiplied by the length of the correspondinglink), over the entire path (x0, x)T and (x + µ, x0)T . The latter being fixed by the treegauge, we see that the information contained in Fx,x+µ(T ) is equivalent to the informationabout Ax,x+µ. We conclude that the values of f on the plaquettes containing tree-links(“along-tree-plaquettes”) can be chosen as independent variables.

Parameterization 3. (“long hydrodynamical invariants”) For a given field configu-ration (φx, Ax,x+µ) on the lattice, we define the lattice analog of hydrodynamical invari-

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ants (2.6):

Rx = |φx| ,vx,x+µ = Ax,x+µ +

αx+µ − αxga(µ)

,

(3.5)

where αx is a phase of φx, i.e. any real number satisfying the equation:

Rx eiαx = φx . (3.6)

Since the phase is defined up to 2nπ, the above quantities are defined only up to a residualgauge transformation

vx,x+µ = vx,x+µ +λx+µ − λxga(µ)

, (3.7)

with the gauge function assuming discrete values only: λx ∈ 2πZ. The value of theelectromagnetic field is equal to the curl of v:

fx;µ,ν =1

a(µ)a(ν)

(a(µ)vx,x+µ + a(ν)vx+µ,x+µ+ν + a(µ)vx+µ+ν,x+ν + a(ν)vx+ν,x

)(3.8)

Sometimes it is useful to exclude from the configuration space the zero-measure set ofconfigurations with vanishing values of R. To parameterize the remaining configurationswe may use ρx = logRx. The space of values of all variables ρ and v describing the initialdata is topologically equivalent to RN+L, where N is the number of sites and L is thenumber of links contained in Σ. We stress however that there is still the residual (discrete)gauge transformation (3.7) acting in this space. The corresponding group is topologicallyequivalent to ZN . The space of initial configurations can be obtained as the quotientwith respect to this group action, i.e. it is equivalent to RL×TN where by TN we denotethe N -dimensional torus. On the classical level this fact has no specific consequences. Wewill however see that in the quantum version of the theory this non-trivial topology willbe responsible for the electric charge quantization.

Parameterization 4. (“short hydrodynamical invariants”) Due to the above transfor-mation properties of vx,x+µ, the following U(1)-valued object is a genuine gauge invariant:

Wx,x+µ = eiga(µ)vx,x+µ =(φxR

−1x

)−1eiga(µ)Ax,x+µ

(φx+µR

−1x+µ

). (3.9)

Obviously, we have

Wx+µ,x = W−1x,x+µ = W ∗

x,x+µ . (3.10)

Passing from v’s to W ’s we partially have lost information about the electromagneticfield. Indeed, due to equation

Wx,x+µ ·Wx+µ,x+µ+ν ·Wx+µ+ν,x+ν ·Wx+ν,x = eiga(µ)a(ν)fx;µ,ν (3.11)

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we are able to reconstruct fx;µ,ν only up to 2nπga(µ)a(ν) . The complete gauge-invariant in-

formation about the field configuration is therefore given by the following set of invari-ants: (Rx,Wx,x+µ, fx;µ,ν), where Rx > 0, |Wx,x+µ| = 1 and f is a closed two-form, fulfill-ing (3.11).

To choose a maximal set of independent parameters among all the above ones, we mayagain use any tree T and to select the set composed of the following invariants:

1) all the values Rx,

2) all the “along-tree” values Wx,x+µ

3) all the fluxes Fx,x+µ(T ) corresponding to all the “off-tree” links.

We see that the compactified (U(1)-valued) degrees of freedom are now described by thequantities Wx,x+µ, whereas the remaining degrees of freedom have trivial topology.

A similar parameterization may be used to describe in a gauge-independent way thespace of Cauchy-data on a three-dimensional layer of the lattice given by {x0 = const}, buthere only space-like µ’s are necessary. The electromagnetic field on space-like plaquettesreduces to the magnetic field: fx;k,l = Bx;k,l and we have:

Wx,x+k ·Wx+k,x+k+l ·Wx+k+l,x+l ·Wx+l,x = eiga2Bx;k,l (3.12)

Since B is a closed form, for any three-dimensional cube the following combination van-ishes identically:

Bx;k,l +Bx;n,k +Bx;l,n +Bx+n;l,k +Bx+l;k,n +Bx+k;n,l ≡ 0 . (3.13)

Again, as independent variables we may take only “along-tree” values of Wx,x+k and only“off-tree” values of the magnetic flux Fx,x+k(T ), for any fixed three-dimensional tree T .

3.4 Reconstruction of the field configuration from invariants

To reconstruct the field configuration (φx, Ax,x+µ) from the “long hydrodynamical invari-ants” (Rx, vx,x+µ) we choose any tree (x0, T ) and put an arbitrary value of Ax,x+µ alongthe tree. We may also choose the phase αx0 at the tree-root in an arbitrary way. Movingalong the tree and using the definition (3.5) of v we may now reconstruct all the phasesαx at all the sites. Then, we reconstruct uniquely the matter field φx from (3.6) and theremaining gauge potentials Ax,x+µ from (3.5).

To reconstruct the field configuration from the “short hydrodynamical invariants” itis, therefore, sufficient to reconstruct the long invariants v. Choose first any values vsatisfying (3.9) and define f as the lattice curl of v. The lattice two-form (f − f) isclosed and integer-valued (more precisely, equation (3.11) implies that it takes values inthe set 2π

ga(µ)a(ν)Z ). Using the lattice version of the Poincare lemma we see that there

exists an integer-valued one-form β on the lattice, such that curlβ = f − f . Hence, theone-form v := v + β fulfills (3.8) and may be taken as a possible representative of longhydrodynamical parameters.

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4 Classical field dynamics

4.1 Lagrangian formulation of dynamics

To obtain the difference equations describing the dynamics of the field on the lattice weuse the following approximation of the Lagrangian (2.1):

Lx = −V(|φx|2)− 12

∑µ

gµµ |(Dφ)x,x+µ|2 − 14

∑µ,ν

gµµgνν (fx;µ,ν)2 , (4.1)

where we sum over positive axes directions only. The action for a given region V ofspace-time is given by

SV =∑

x∈Vτa3Lx , (4.2)

where we sum only over the internal sites. As we already mentioned, the fields on theboundary sites and links have to be fixed. They are used as the boundary conditionsfor the dynamics. The dynamics is deduced from the above action and is given by thefollowing system of second-order difference equations:

0 =∂S

∂φx= −1

2τa3

2V′ · φx −

∑

µ

gµµ

a(µ)(Dφ)x,x+µ

,

0 =∂S

∂Ax,x+µ= −gµµτa3

(g Im

(φx (Dφ)x,x+µ

)+∑

ν

gνν

a(ν)fx;µ,ν

).

(4.3)

Introducing the canonical momenta, analogous to (2.8),

πµx = −gµµ(Dφ)x,x+µ ,

pµνx = −gµµgννfx;µ,ν ,

(4.4)

the same equations may be written as a system of first-order difference equations

−2V′ · φx =∑

µ

1a(µ)

πµx ,

−g Im(φx π

µx

)=

∑

ν

1a(ν)

pµνx .

(4.5)

13

4.2 Canonical momenta

Initial data for the theory consist of field configurations and the corresponding momenta ina single three-dimensional lattice layer Σt, given by {x0 = t}. Because momenta (4.4) arealways attached to the time-like lattice links and plaquettes, we may choose between thefuture-directed and the past-directed momenta. In the first case the complete Cauchy dataare given by the collection (φx, Ax,x+k, π

0x, p

0kx ). To keep the future orientation of the mo-

menta, we will describe the Cauchy data in the second case by (φx, Ax,x+k,−π−0x ,−p−0k

x ).We will call the first collection “the Cauchy data on Σ+

t ” and the second one “on Σ−t ”.We may imagine Σ+

t (respectively Σ−t ) as the hypersurface obtained from Σt by a tinytranslation in the positive (resp. negative) direction of the time axis.

To simplify the notation we will denote

π+x := π0

x

π−x := −π−0x

D+x,x+k

:= −p0kx = −fx;0,k

D−x,x+k

:= p−0kx = fx;−0,k

(4.6)

where by D we denote the electric components of the electromagnetic field (the magneticcomponents are already denoted by B). Notice that, similarly as in the continuous versionof the theory, the momentum canonically conjugate to the gauge potential Ak is given byminus the electric induction field. This means that the momentum conjugate to Ax,x+k

on Σ+ (resp. Σ−) is given by −D+x,x+k

(resp. −D−x,x+k

).There is, however, an additional factor a3, which appears in the canonical structure of

the Cauchy data. Indeed, in the continuous version of the theory the canonical structureis given by the following Poisson bracket:

{Re π(x),Re φ(y)

}= δ(3)(x− y) (4.7)

and similar formulae for the remaining degrees of freedom. Integrating the above formulaover a domain V ⊂ Σ with respect to the variable x we obtain

{ ∫

VRe π(x)d3x,Re φ(y)

}= δV,y :=

{1, when y ∈ V ;0, otherwise .

(4.8)

In the lattice version of the theory the discretized observables represent their mean valuesover elementary cubes. Hence, we should choose as V such an elementary cube and replacethe integral by a3Re π. This way we obtain the canonical Poisson bracket:

{a3Re πx,Re φy

}= δx,y . (4.9)

We conclude that the momentum canonically conjugate to φx is equal to a3πx, and themomentum canonically conjugate to Ax,x+k is equal to −a3Dx,x+k.

14

4.3 Initial value problem

The transition from Σ−t to Σ+t will be called potential evolution. The transition from Σ+

t

to Σ−t+τ will be called kinetic evolution. The global evolution of the field is given by thesuccessive composition of the potential and kinetic evolution operators.

As a result of the potential evolution, the field configurations (φx, Ax,x+k) remain un-changed and the momenta change their value from (π−x , D

−x,x+k

) to (π+x, D+

x,x+k) according

to formulae (4.5). We have therefore:

π+x = π−x − 2τV′ · φx +

τ

a

∑

k

(Dφ)x,x+k , (4.10)

and

D+x,x+k

= D−x,x+k

+ gτ Im(φx (Dφ)x,x+k

)+τ

a

∑

l

Bx;k,l , (4.11)

During the kinetic evolution the momenta are transported parallelly:

π−x+0 = e−igτAx,x+0 π+

x , (4.12)

and

D−x+0,x+0+k

= D+x,x+k

, (4.13)

whereas the configurations undergo the linear evolution according to formulae (4.4). Thisimplies for the matter field the following evolution:

eigτAx,x+0 φx+0 = φx + τπ+x . (4.14)

Hence, the kinetic evolution is given uniquely up to a gauge transformation. Indeed, fixingin an arbitrary way the value of the gauge field on time-like links we may derive uniquelythe final data from the initial data. Putting e.g. Ax,x+0 = 0 we obtain from (4.4):

φx+0 = φx + τπ+x , (4.15)

Ax+0,x+0+k = Ax,x+k − τD+x,x+k

, (4.16)

whereas the momenta remain constant due to (4.12) and (4.13).We stress that both the potential and the kinetic evolution are local: to find the final

data in a bounded portion V of the lattice we do not need to know the entire initial dataon Σ, but only a portion of it, contained in the discrete causal shadow of V .

On quantum level the potential evolution will be generated by the potential part ofthe action (i.e. the sum of all the terms assigned to the lattice sites and to the purelyspace-like lattice links and plaquettes), whereas the kinetic evolution will be generatedby the kinetic part of the action (i.e. the sum of the terms assigned to the time-like linksand plaquettes). In the τ → 0 limit we obtain, due to the Trotter formula, the continuousevolution generated by the sum of the potential and the kinetic energy.

15

4.4 Gauge-invariant description of initial data

Due to the gauge properties of φ and π, their combination πφ is gauge invariant. Dividingit into the real part K and the imaginary part M we observe that the later has to fulfilla constraint analogous to (2.11), implied by the zero component of (4.5):

1g

1a

∑

k

D+x,x+k

−M+

x = 0 =1g

1a

∑

k

D−x,x+k

−M−

x , (4.17)

where we denote

π+x φx =: K+

x + iM+x

π−x φx =: K−x + iM−x .

(4.18)

Similarly as in the continuous theory, we may reduce the phase space of Cauchy dataon every Σt with respect to these constraints. As a result we obtain a formula analo-gous to (2.12) with the continuous divergence operator replaced by its lattice analog. Weconclude that K (with the index “+” if we are on Σ+ and “−” if we are on Σ−) is the mo-mentum canonically conjugate to ρ = logR, whereas the electric induction field D is themomentum canonically conjugate to the long hydrodynamical variable v. The vanishingleft- (resp. right-)hand-side of (4.17) have to be considered as the momenta canonicallyconjugate to the (completely arbitrary) phase of the matter field. Knowing (R, v,K,D)on Σ we may first reconstruct the configuration (φ,A) (uniquely up to a gauge trans-formation). Then, we obtain the missing information M about the momentum π fromthe lattice divergence of the electric induction field D, according to (4.17). This way wereconstruct (uniquely up to a gauge transformation) the complete initial data (φ,A, π,D).

Splitting the matter field into 2 degrees of freedom corresponding to its real andimaginary part: φ = φ1 + iφ2, the corresponding momentum decomposes accordinglyinto π = π1 + iπ2. Thus

K = φ1π1 + φ2π2 ,

M = φ1π2 − φ2π1 .

(4.19)

are the generators of the two-dimensional dilatations and rotations respectively.

4.5 Time evolution in terms of gauge invariants

Both the potential and the kinetic evolution may be rewritten in terms of the invariants.During the potential evolution the configurations (R,W,B) remain unchanged whereas

16

the following equations for the momenta may be immediately deduced from (4.10), (4.11)and the definition (4.18) of K:

K+x

= K−x− 2τV′

(R2x

)·R2

x −τ

a2

∑

k

(R2x −RxRx+kReWx,x+k

),

D+x,x+k

= D−x,x+k

+ gτ

aRxRx+k ImWx,x+k +

τ

a

∑

l

Bx;k,l .

(4.20)

The last equation is the lattice version of the Maxwell equation D = −j + curlB.The simplest way to rewrite the kinetic evolution in terms of invariants is to fix the

temporal gauge Ax,x+0 = 0. With such a choice the kinetic evolution of the matter fieldover each time-like link becomes a free evolution of the two degrees of freedom describedby φ. The evolution may be thought of as generated by the free Hamiltonian a3H, where

H =12ππ =

12R−2(K2 +M2) , (4.21)

is the Hamiltonian density and the factor a3 comes from the integration over an elementarycube. It is obvious that H remains constant during the kinetic evolution. This implies thatits value on both ends of the link is the same. Also the “angular momentum” M remainsconstant in agreement with formula (4.17) (remember that M is not an independentvariable but has to be treated as the divergence of the electric induction field).

Calculating the Poisson bracket of various quantities with the above Hamiltonian weobtain their evolution along the link (x, x+ 0). Hence,

K = {K, a3H} = 2H (4.22)

implies

K−x+0 = K+

x+ 2τH+

x. (4.23)

Moreover,

(R2)· = {R2, a3H} = 2 (φ1π1 + φ2π2) = 2K (4.24)

implies the evolution of R, compatible with (4.15):

R2x+0 = |φx+0|2 =

=∣∣∣φx + τπ+

x

∣∣∣2

=

= R2x + 2τK+

x + 2τ 2H+x .

(4.25)

17

The lattice curl of equation (4.16) implies the evolution of the magnetic field, analogousto the Maxwell equation B = −curlD:

Bx+0;kl = Bx;k,l −τ

a

(D+x,x+k

+D+x+k,x+k+l

+D+x+k+l,x+l

+D+x+l,x

). (4.26)

To derive the kinetic evolution of W let us observe, that the definition of D implies:

Wx+0,x+0+k = Wx+0,xe−igτaD+

x,x+kWx,x+kWx+k,x+0+k . (4.27)

Using the evolution of φ written in polar coordinates

φx+0 = φx + τπ+x

=

= φx

[1 + τR−2

x

(K+x

+ iM+x

)],

(4.28)

we may express Wx,x+0 in terms of Cauchy data on Σ+ or on Σ−:

Wx,x+0 =(φxR

−1x

)−1φx+0R

−1x+0 =

= Rx

[1 + τR−2

x

(K+x

+ iM+x

)] (R2x + 2τK+

x+ 2τ 2H+

x

)− 12 =

=(R2x+0 − 2τK−

x+0 + 2τ 2H−x+0

)− 12[1− τR−2

x+0

(K−x+0 − iM−

x+0

)]Rx+0 .

(4.29)

Finally, substituting (4.29) to (4.27) we easily find the evolution of the “short” hydrody-namical variables:

Wx+0,x+0+k =(R2x + 2τK+

x+ 2τ 2H+

x

)− 12[1 + τR−2

x

(K+x− iM+

x

)]×

× Rxe−igτaD+

x,x+k Wx,x+k ×Rx+k

×[1 + τR−2

x+k

(K+x+k

+ iM+x+k

)] (R2x+k + 2τK+

x+k+ 2τ 2H+

x+k

)− 12 .

(4.30)

5 Quantum kinematics

5.1 Quantum states of the field

Let us begin with the naive Schrodinger representation of a quantum state Ψ as a wavefunction of field configurations:

Ψ = Ψ({φx}, {Ax,x+k}) (5.1)

18

on our finite three-dimensional lattice Σ. The momenta canonically conjugate are repre-sented by derivatives with respect to configurations:

a3(Re πx) =h

i

∂

∂(Re φx),

a3(Im πx) =h

i

∂

∂(Im φx),

(5.2)

(similarly as in the classical case, the factor a3 appears because of the integration ofthe densities of the above momenta over elementary cubes of the lattice). The abovedefinitions may again be shortened as follows:

a3πx = 2h

i

∂

∂φx. (5.3)

Similarly,

−a3Dx,x+k =h

i

∂

∂Ax,x+k. (5.4)

This implies that the generator of two-dimensional rotations M = φ1π2 − φ2π1 (there isno ordering problem here!) is represented as the derivative with respect to the phase αxof φx:

a3Mx =h

i

∂

∂αx. (5.5)

To distinguish between the momenta on Σ+ and Σ− we will mark them, when necessary,with + or −, respectively.

The physically meaningful quantum states have, however, to fulfill the Gauss con-straint (4.17). Written in terms of the above momenta, the constraint implies the vanish-ing of the following operator:

Gx = − ∂

∂αx− 1ga

∑

k

∂

∂Ax,x+k. (5.6)

But ΣλxGx is the generator of the gauge transformation (3.3). Wave functions which areannihilated by Gx are, therefore, those which are constant on gauge orbits. They cannot be square-integrable with respect to all the gauge parameters because gauge orbitsare not compact. To define the quantized version of our theory we have to organize suchfunctions in a Hilbert space. For this purpose we have to define a scalar product in thespace of gauge-invariant wave functions. There have been many proposals how to definesuch a scalar product (cf. [1], [2], [16] and [17]). In this paper we shall use the productdefined in [19] and [20] on geometrical grounds. However, we are not going to imposethis definition a priori, but will derive it uniquely from the representation theorem forthe algebra of observables.

19

5.2 Algebra of quantum observables

In the naive Schrodinger representation used above, quantum observables are those oper-ators, which are gauge-invariant. The gauge-invariant functions of configurations belongto this class. They can be used as multiplication operators acting on gauge-invariantfunctions. This way we define quantum observables (Rx,Wx,x+k, Bx;k,l), where all Rx areself-adjoint and positive, Wx,x+k are unitary and Bx;k,l are self-adjoint. All these operatorscommute with each other. They are not independent because they obviously fulfill theconstraints (3.12) and (3.13).

Among the momenta, the electric induction field operator Dx,x+k given by (5.4) andthe generator of the two-dimensional dilatation

Kx :=12

(φ1π1 + π1φ1 + φ2π2 + π2φ2) (5.7)

(symmetrized version of the classical quantity (4.19)) produce a gauge-invariant functionwhen applied to a gauge-invariant function. Hence, they are also observables. Denote byOΣ the algebra generated by the above operators. The following commutation relationsbetween the generators of OΣ can be immediately checked:

[Rx, a

3Kx

]= ihRx ,

[Wx,x+k,−a3Dx,x+k

]= −hga Wx,x+k ,

[Bx;k,l,−a3Dx,x+k

]=

ih

a1 ,

(5.8)

with all the remaining commutators vanishing.

Definition 1 The charge Qx ∈ OΣ at a lattice site x is defined as the lattice divergenceof the electric induction field:

Qx = a3divxD = a2∑

k

Dx,x+k = ga3Mx . (5.9)

The commutation relation

[Wx,x+k,1gQx] = h Wx,x+k , (5.10)

follows immediately from the above definition of Q and from (5.8).

Lemma 1 The operator Ux := e2πieQx commutes with all the observables in OΣ.

20

Proof. The operator Qx is built of the operators D. Therefore, it commutes with allthe observables Ry and Ky. Hence, we have to check the commutation of U with W and B.The commutator [divxD,Bx;k,l] vanishes, because it is a sum of the term [Dx,x+k, Bx;k,l] =iha4 and the term [Dx,x+l, Bx;k,l] = − ih

a4 (B is antisymmetric in k, l). To check the commu-tation with W consider a one-parameter family of operators

W λx,x+k = e−iλa

3divxDWx,x+keiλa3divxD . (5.11)

Its derivative over λ equals

∂W λx,x+k

∂λ= e−iλa

3divxD[−i a3divxD Wx,x+k + i Wx,x+k a

3divxD]eiλa

3divxD (5.12)

and contains the commutator of a3divD and W equal to

a2[Dx,x+k,Wx,x+k] = −ghWx,x+k . (5.13)

Hence,

∂W λx,x+k

∂λ= ighW λ

x,x+k , (5.14)

and consequently

W λx,x+k = eighλW 0

x,x+k , (5.15)

thus, for λn = 2πgh·n with n ∈ Z we have W λn = W 0 = W . This implies, that the operator

eiλ1a3divxD = e2πighQx (5.16)

commutes with all the observables, which ends the proof of the lemma.

5.3 Existence and uniqueness of irreducible representation

Algebra OΣ could be also defined without any use of the Schrodinger representation. Forthis purpose, classical gauge-invariant observables (R,W,B,D,K) can be taken, withtheir Poisson brackets replaced by commutators. Quantization of such a structure meansa construction of its irreducible Hilbert-space representation.

The Schrodinger representation in the space of all L2 functions of field configurationsφx and Ax,x+k is not irreducible, since there is a non-trivial, unitary operator Ux whichcommutes with all the observables. For irreducible representations, this commutativitywould imply

Ux = e2πiθx · 1 . (5.17)

21

The electric charge quantization would be an immediate consequence of the above condi-tion. Indeed, the spectrum of Qx covers the set {gh ·n+ θx, n ∈ Z}. Hence, e := gh playsthe role of the elementary charge.

There are however, non-equivalent irreducible representations, corresponding to differ-ent values of the angle θx. To avoid non-symmetry of the charge spectrum we will alwaysimpose condition

θx ≡ 0 , (5.18)

which implies already the uniqueness of the representation.To prove this statement let us first construct a representation by introducing the

following scalar product:

Definition 2 If Ψ and Ξ are gauge-invariant functions of field configurations φx andAx,x+k on Σ, their scalar product is given by the formula:

(Ψ|Ξ) =1

2π

∫ΨΞ

∏

(x,x+k)6∈TdAx,x+k

∏x

dφxdφx , (5.19)

where T is any three-dimensional tree in Σ.

It is easy to see that the above definition does not depend upon the specific choice ofthe tree T and the choice of a gauge on it (i.e. the choice of the values of the “along-treegauge potentials”). In fact, we integrate over the space of gauge-orbits, parameterized by“tree-parameters”. Moreover, we use the invariance of the induced measure with respectto a particular choice of the “tree-gauge”.

The Hilbert space generated by the above scalar product will be denoted HΣ.Elements of OΣ act in a natural way as operators in HΣ. Due to constraint (5.6),

the operator Qx reduces to “two-dimensional angular momentum” operator (5.5) whosespectrum is {gh · n, n ∈ Z}. Hence, the condition (5.18) has been fulfilled.

Theorem 1 For a finite three-dimensional lattice Σ, there exists only one (up to unitaryequivalence) representation of the algebra OΣ, which fulfills the condition (5.18).

Proof. Let us choose any three-dimensional tree T in Σ. To each site x different fromthe root x0 we assign the following unitary operator:

Wx(T ) :=∏

(y,y+k)∈(x0,x)T

Wy,y+k . (5.20)

Take the following collection OΣ(T ) of operators:

1) (ρx, Kx) at all the lattice sites

22

2) (Wx(T ), 1gQx) at all the lattice sites different from the root x0

3) (Fx,x+k(T ), Dx,x+k) on all the off-tree links.

It is easy to see that OΣ(T ) may be chosen as the minimal Lie-subalgebra generating theentire algebra OΣ and that the commutation relations (5.8) are equivalent to the canonicalcommutation relations between pairs of operators 1), 2) and 3):

[ρx, a

3Kx

]= ih1 ,

[Wx(T ),

1gQx

]= −hWx ,

[1aFx,x+k(T ), a3Dx,x+k

]= −ih1 ,

(5.21)

with all the remaining commutators vanishing. All the representations of such a Heisen-berg algebra are equivalent to the Schrodinger representation in terms of the wave func-tions “Ψ” depending on the parameters (ρx,Wx(T ), 1

aFx,x+k(T )) (the so called θ-repre-

sentations of Qx are excluded by the condition (5.18)). The wave functions are square-integrable with respect to the Lebesgue measure on the first and the last group of pa-rameters and with respect to the standard measure (Wx)−1dWx on the circle. The mo-menta (a3Kx,

1gQx, a

3Dx,x+k) are represented as differentiations with respect to the abovevariables.

To see that the above Schrodinger representation is equivalent to the one constructedbefore, it is sufficient to observe that in the tree-gauge A ≡ 0 on T we have

φx = eρx Wx(T )arg(φx0) (5.22)

outside of the tree root and

Ax,x+k =1aFx,x+k(T ) (5.23)

on the off-tree links. We stress that the phase arg(φx0) of φx0 at the tree root is completelyarbitrary and the corresponding momentum 1

gQx0 is not an independent variable: its

value is equal to the value of the total electric charge contained in Σ (determined bythe boundary condition which we imposed) minus the sum of the remaining charges 1

gQx

outside of the tree root.Hence, the formula

Ψ =∏x

e−ρxΨ (5.24)

defines a square-integrable function with respect to the Lebesgue measure dφdφdA whichwe used in the definition (5.19), i.e. an element of our Hilbert space HΣ. The transfor-mation is obviously unitary because the integration over arg(φx0) in (5.19) kills the factor2π. This ends the proof.

23

5.4 Unbounded lattice

There is a natural inclusion OΣ1 ⊂ OΣ2 when Σ1 ⊂ Σ2. Hence, the algebra OΣ for thecomplete, unbounded lattice Σ = Z3 can be immediately obtained as the inductive limitof the algebras corresponding to bounded Σ’s. Unfortunately, the representation of thecommutation relations (5.8) for the unbounded lattice can not be unique, because therepresentation of the Heisenberg commutation relations (5.21) is not unique for infinitelymany degrees of freedom. The theory splits, therefore, into separate, non interactingsectors. Different sectors of the theory may be constructed e.g. from different vacuumstates.

The space of all the (mixed) states σΣ1 for a bounded lattice Σ1 can be treated as adual of OΣ1 . The inclusion of OΣ1 into OΣ2 defines in a unique way the projection

PΣ1,Σ2 : σΣ2 → σΣ1 . (5.25)

The space of quantum states σΣ on the unbounded lattice can thus be constructed asa projective limit of the spaces corresponding to bounded Σ’s. Choosing a single states ∈ σΣ as a vacuum, enables us to reconstruct the entire sector of the theory via thestandard Gelfand-Naimark-Segal procedure. The physically correct choice of the vacuumwill be discussed in the sequel. We stress, however, that each Hilbert space obtained thisway corresponds to a fixed value of the total electric charge and the “superselection rules”are automatically satisfied. Hence, for describing the interaction of charged particles, thenon-vacuum sectors will be also important.

6 Quantum dynamics

6.1 The Feynman path integral in terms of gauge invariants

To define the evolution of the quantum system defined above, we begin with a discreteversion of formula (1.2). Both the left and the right-hand-sides are functions of thefields (φz, Az,z+k) on the surface Σfin. To compute the value of the right-hand-side wehave to integrate over initial and intermediate configurations, with the final configura-tions (φz, Az,z+k) being fixed. But, since the integrand is constant on gauge-orbits, theintegration over all intermediate configurations will always be ill defined. Similarly as inthe definition of the Hilbert space HΣ, we have to define the integral in such a way thateach gauge orbit is taken only once.

Definition 3 The quantum evolution on the lattice is given by

U(Σfin,Σinit)Ψinit =1N

∫eihSΨinit(φx, Ax,x+k)

∏

tinit<y0<tfin

dφydφy ×

24

× ∏

(y,y+µ)6∈TdAy,y+µ

∏

x∈Σinit

dφxdφx∏

(x,x+k)6∈TdAx,x+k ,

(6.1)

where T is any four-dimensional tree, covering the intermediate region {tinit < y0 < tfin}between Σinit and Σfin, together with the initial surface Σinit, and the corresponding tree-gauge has been chosen in an arbitrary way. The action S corresponding to the region{tinit < y0 < tfin} is given by formula (4.2).

Similarly as in the case of formula (5.19) it is easy to check that the result does notdepend upon the specific choice of the tree and the choice of the gauge. Moreover, becauseof the gauge invariance of the initial wave function and the action S, the result is a gauge-invariant function of final parameters, i.e. it describes a physical quantum state on Σfin.Hence, we have defined an operator U(Σfin,Σinit) from HΣinit to HΣfin . We will show in thesequel, that the normalization factor N may be chosen in such a way, that U is unitary.

6.2 Factorization of the evolution

The simplest way to compute the value of the integral (6.1) consists in choosing the four-dimensional tree T which is composed of a three-dimensional tree Tinit in Σinit and all thetime-like links (y, y+ 0). Moreover, we choose a zero temporal gauge (for the moment wedo not need to specify the value of the gauge on Tinit). First, let us perform the integrationover the intermediate configurations. This way we obtain a kernel depending upon theinitial and the final configurations. But the action S is a sum of kinetic terms

Skin =a3

2τ

∑y

|φy+0 − φy|2 +a3

2τ

∑

y,k

(Ay+0,y+0+k − Ay,y+k)2 , (6.2)

assigned to each elementary time-interval of the discretized time axis, and potential terms

Spot = −τa3∑y

V(|φy|2)− 12τa3

∑

y,k

|(Dφ)y,y+k|2 −14τa3

∑

y,k,l

(By;k,l)2 , (6.3)

assigned to each surface {y0 = const}. Hence, the kernel U(Σfin,Σinit) is equal to thesuccessive superposition of the kinetic kernels

Ukin =1NeihSkin (6.4)

and the potential kernels

Upot = eihSpot . (6.5)

25

Observe that the kinetic kernel is equal to the resolvent kernel corresponding to thelinear Schrodinger equation:

U(t+τ,t)(y, x) =1√

2πhτeihτ

(x−y)2. (6.6)

The kinetic evolution consists, therefore, in the free Schrodinger evolution during thetime interval τ of all the degrees of freedom (φy, Ay,y+k) on Σ. Such an evolution kernel isobviously unitary when applied to any square-integrable function of initial configurations(φx, Ax,x+k). Applying this kernel means integrating over all the parameters (φx, Ax,x+k).The normalization factor equals (2πhτ)−

12 for each degree of freedom.

In our case we are going to apply the kernel to gauge-invariant wave functions only.This means that, according to formula (6.1), we are going to integrate over the off-tree parameters only. We stress, however, that this difference does not produce anyproblem, because both integrals give the same value when applied to a gauge-invariantwave function. This is due to the oscillatory (Fresnel-like) character of the kernel. Theintegration of Ukin with respect to the gauge parameters over each gauge orbit will giveus the value 1, so only the integration over the space of orbits remains. The situation issimilar to the following example: we may apply the three-dimensional free Schrodingerevolution to wave functions which do not depend upon the z-variable. The integrationover dz does not produce any difficulty due to the oscillatory character of the Fresnelkernel.

The potential evolution, consisting in multiplying the wave function by Upot, is obvi-ously unitary because |Upot| = 1.

6.3 Quantum evolution in the Heisenberg picture

It is very instructive to reconsider the above evolution in the Heisenberg picture, where thequantum state remains constant in time and the quantum observables evolve accordingto the formula O → O := U−1OU . It is obvious that the potential evolution leavesthe configurations (R,W,B) invariant. The evolution of momenta (K,D) may be easilyobtained, according to formulae:

K+ Ψ = U−1potK

−Upot Ψ ,

D+ Ψ = U−1potD

−Upot Ψ .

(6.7)

This way we obtain the following formulae, analogous to classical equations (4.20):

R+x

= R−x,

W+x,x+k

= W−x,x+k

,

26

B+x;k,l

= B−x;k,l

,

K+x

= K−x− 2τV′

(R2x

)·R2

x −τ

a2

∑

k

R2

x −RxRx+k

Wx,x+k

+W−1x,x+k

2

,

D+x,x+k

= D−x,x+k

+ gτ

aRxRx+k

Wx,x+k −W−1x,x+k

2i+τ

a

∑

l

Bx;k,l .

(6.8)

In a similar way we may obtain the potential evolution of the electric charge M givenby (5.5):

M+x = U−1

potM−x Upot =

= M−x +

i

h(M−

x Spot) =

= M−x +

τ

a2Rx

∑

k

Rx+k

Wx,x+k −W−1x,x+k

2i.

(6.9)

This equation is the lattice version of the continuity equation: the value of the electriccharge M changes only by the divergence of the electric current carried by the matterfield.

The corresponding formulae for the kinetic evolution will be derived in the next Sec-tion. Here we give only the final results. To simplify our notation we introduce thefollowing quantity, analogous to the classical Wx,x+0, defined in (4.29):

Wx,x+0 = W−1

x+0,x =(φxR−1x

)−1φx+0R

−1x+0 =

= Rx

[1 + τR−1

x

(K+x

+ iM+x

)R−1x

] (R2x + 2τK+

x+ 2τ 2H+

x

)− 12 .

(6.10)

It will be proved in the next section, that the same quantity may be expressed in termsof the Cauchy data on Σ−:

Wx,x+0 =

(R2x+0 − 2τK−

x+0 + 2τ 2H−x+0

)− 12 ×

×[1− τR−1

x+0

(K−x+0 − iM−

x+0

)R−1x+0

]Rx+0 .

(6.11)

Furthermore, we denote as usually:

Mx =1ga

∑

k

Dx,x+k (6.12)

27

and

Hx =12R−1x

(K2x +M2

x

)R−1x , (6.13)

on both Σ+ and Σ−.The kinetic evolution now reads:

Rx+0 =(R2x + 2τK+

x + 2τ 2H+x

) 12 ,

Wx+0,x+0+k = Wx+0,xe−iga τ2D+

x,x+kWx,x+ke−iga τ2D+

x,x+kWx+k,x+0+k ,

Bx+0;k,l = Bx;k,l −τ

a

(D+x,x+k

+D+x+k,x+k+l

+D+x+k+l,x+l

+D+x+l,x

),

K−x+0 = K+

x+ 2τH+

x,

D−x+0,x+0+k

= D+x,x+k

.

(6.14)

We stress that the quantum evolution is given by the same equations as in the classicalcase (formulae (4.25) – (4.30)). However, the evolution for W is given by the product ofnon-commuting operators. Changing their order changes the result completely. In (6.14)we have chosen an ordering, which formally reproduces the classical equation (4.30).

Field equations (6.8) and (6.14) contain the complete set of Maxwell equations. Infact, divB = 0 is an identity. The kinetic evolution reproduces the equation B = −curlD.The potential evolution reproduces D = −j + curlB, and the last equation divD = q isthe definition of the electric charge density.

We stress that, similarly as in the classical version of the theory, the quantum dynamicsis causal. Indeed, the evolution of a momentum at the site x depends on variables at thenearest neighbors of x only. This way, knowing the quantum observables in a boundedregion of Σ, we can compute the time evolved operators in an entire future causal shadowof this region.

6.4 Self-consistency of the evolution

In the evolution formulae (6.8) and (6.14) we have applied a potentially dangerous op-eration, the square root, to a time-dependent operator. To prove, that our formulae areformally correct we have to show, that this operator is non-negative for all times.

Theorem 2 The operator R2 + 2τK + 2τ 2H is a positive, self-adjoint operator for allvalues of τ .

28

Proof. The operator in question is equal to the square of the modulus of the followingoperator: N = R + τR−1(K + iM + ih). Indeed, we have:

N †N =[R + τ(K − iM − ih)R−1

] [R + τR−1(K + iM + ih)

]=

= R2 + 2τK + τ 2(K − iM − ih)R−2(K + iM + ih) == R2 + 2τK + 2τ 2H .

(6.15)

We also have the following

Theorem 3 The operator W , given by the right-hand side of (6.14) remains unitary if Wwas unitary.

Proof. Unitarity of the evolvedWx,x+k follows from formula (6.14), if we prove unitarityof Wx,x+0:

W †x,x+0Wx,x+0 =

=(R2x + 2τK+

x+ 2τ 2H+

x

)− 12[Rx + τR−1

x

(K+x− iM+

x

)]×

×[Rx + τ

(K+x

+ iM+x

)R−1x

] (R2x + 2τK+

x+ 2τ 2H+

x

)− 12 =

=(R2x + 2τK+

x+ 2τ 2H+

x

)− 12[R2x + 2τK+

x+ τ 2R−1

x

((K+

x)2 + (M+

x)2)R−1x

]×

×(R2x + 2τK+

x+ 2τ 2H+

x

)− 12 =

= 1 .(6.16)

This proves, that equations (6.8) and (6.14) define a self-consistent quantum dynamicsof our algebra of observables.

6.5 Boundary conditions

Quantum evolution of Heisenberg operators described by formulae (6.8) – (6.14) is causal.This statement means, that we may solve formally the initial value problem for quantumoperators in the same way, as we did for the classical gauge-invariants. This way, each localquantum observable may be represented as a combination of the observables contained inits discrete “past light cone”. In the above statement the “speed of light” is understood ina purely combinatorial sense, corresponding to the structure of dynamical equations (6.8)

29

– (6.14): one lattice spacing in the direction of time corresponds to at most one latticespacing in the direction of space.

The above procedure cannot, however, be continued if the above “light cone” hits theboundary of our finite lattice Σ. Similarly as in classical case, to continue the solution, onehas to impose appropriate boundary conditions. These conditions are of purely classical(“c-number”) character. They will be imposed on a three dimensional surface B definedas Cartesian product T × ∂Σ of the time axis T and the two dimensional boundary of Σ.The observables assigned to elements of the lattice B are thus non-dynamical and purelyclassical.

The boundary data which we have to fix on B are Rx and Wx,x+0 at all the nodes x ∈ Band Dx,x+k at all the links (x, x+ k) such that both x and x+ k belong to B.

To prove that the dynamics is now well defined we first analyze the potential evolution.The configurations R, W and B do not change during the potential evolution (6.8).To determine the momentum K+

xwe need K−

x, Rx+k in the space-like neighbors of x

and Wx,x+k on all the space-like links starting at x. The boundary value of Ry beingfixed, the evolution of K is, therefore, well defined. The potential evolution of the electricinduction D is determined by the “internal” (i. e. dynamical) data.

The kinetic evolution of internal momenta Kx and Dx,x+k uses only internal quantities.The same holds for Rx. To determine Wx+0,x+0+k we need Dx,x+k as well as both Wx,x+0and Wx+k,x+0+k. At the internal lattice nodes we may compute Wx,x+0 in terms of Rx,Kx and Mx = 1

ga

∑kDx,x+k but we have to fix its value at the boundary. Finally, to be

able to determine the magnetic field B on a future internal plaquette we need to knowthe electric induction on all the boundary of the respective plaquette. Hence, we have touse the value D inside the boundary B.

This ends the consistency proof of the field dynamics.

7 Kinetic evolution of observables

7.1 Kinetic evolution of matter fields

To calculate the kinetic evolution of our observables, we take first the free Schrodinger evo-lution of all the degrees of freedom (ψ,A, π,D) in the temporal gauge Ax,x+0 = 0. Finally,we will use these results to compute the evolution of their gauge-invariant combinations(R,W,B,K,D). This way we obtain results, which are already gauge independent.

We stress that the “free evolution of φ and A”, which we use in this Section, is onlya convenient way of calculation and has no physical significance. In fact, in our approachthere is no way to give any reasonable meaning to quantities φ and A on the quantumlevel.

The free Schrodinger evolution of the two degrees of freedom (Re φx, Im φx) carried byφx may be rewritten in terms of the operators (R,F,K,M), where φ = F ·R is the polar

30

decomposition of the normal operator φ = φ1 + iφ2. These operators fulfill the followingcommutation relations

[R,K] = iha−3R ,

[F,M ] = −ha−3F ,

(7.1)

all other commutators are equal zero.The free Hamiltonian (equal to the two-dimensional Laplacian in the variable φ) may

be written as follows:

H = − h2

2∆φ =

12

(π21 + π2

2) =12R−1(K2 +M2)R−1 , (7.2)

a3H being the generator of the time evolution. The evolution of the field operators in theHeisenberg picture is given by equations:

R =i

h[a3H,R] = R−1(K + i

h

2a3)

F =i

h[a3H,F ] = iR−2(M − h

2a3)F

K =i

h[a3H,K] = 2H

M =i

h[a3H,M ] = 0 .

(7.3)

The Hamiltonian H and the angular momentum M are constant in time. Hence, theevolution of K is linear

K(t) = K(0) + 2H(0)t . (7.4)

Moreover, for C = R2 we have

C =i

h[a3H,C] = 2K , (7.5)

which implies

C(t) = C(0) + 2K(0)t+ 2H(0)t2 . (7.6)

We already proved that the above operator remains always positive. To find R(t) we haveto take its square root.

To find the evolution of F let us consider the product RF . We have

(RF ) =i

h[a3H,RF ] = R−1(K + iM)F . (7.7)

31

Observe that the right-hand-side commutes with the Hamiltonian and, therefore, is con-stant in time. Hence, we have

R(t)F (t) =[R(0) +R(0)−1 (K(0) + iM(0)) t

]F (0) . (7.8)

Summarizing, after the time τ we have

H(τ) = H(0)K(τ) = K(0) + 2τH(0)

R(τ) =(R(0)2 + 2τK(0) + 2τ 2H(0)

) 12

M(τ) = M(0)

F (τ) =(R(0)2 + 2τK(0) + 2τ 2H(0)

)− 12[R(0) + τR(0)−1 (K(0) + iM(0))

]F (0)

(7.9)

7.2 Kinetic evolution of gauge fields

In our temporal gauge each degree of freedom Ax,x+k undergoes separately the one-di-mensional, free Schrodinger evolution and its canonical momentum D = − h

ia3∂∂A

remainsconstant. Hence, after time τ we have

A(τ) = A(0) +hτ

ia3

∂

∂A= A(0)− τD(0) . (7.10)

Taking the lattice curl of both sides we have

B(τ) = B(0)− τ curlD(0) , (7.11)

reproducing the free Maxwell equation.

7.3 Kinetic evolution of physical observables

Using the evolution of gauge-dependent fields, we may finally find the evolution of thegauge-invariant quantity

Wx,x+k(t) = eigaAx,x+k(t)Fx+k(t)F−1x

(t) (7.12)

(no ordering problem because the operators on the right-hand side commute). Moreover,according to (6.10), we have

Wx,x+0 =(φxR

−1x

)−1φx+0R

−1x+0 = F−1

xFx+0 . (7.13)

32

Using the previous results it is easy to compute:

Wx+0,x+0+k = F−1x+0e

igaAx+0,x+0+kFx+0+k =

=(F−1x+0Fx

)F−1xe−iga τ2D+

x,x+keigaAx,x+ke−iga τ2D+

x,x+kFx+k

(F−1x+k

Fx+0+k

)=

= W−1x,x+0e

−iga τ2D+x,x+kWx,x+ke

−iga τ2D+x,x+kWx+k,x+0+k ,

(7.14)

which immediately implies (6.14).

8 Formulation in terms of configuration variables

8.1 Algebra of observables

The entire theory can also be rewritten in terms of field configurations only, without anyuse of field momenta. To keep a complete set of initial data, we have to replace theinformation which was contained in the momenta by the information carried by the fieldconfiguration on an adjacent Σ. This way, we can describe the observable algebra by thefield configurations over two consecutive space-like hyperplanes Σt and Σt+0, together withall the time-like links and plaquettes between them. In this formulation it is convenientto use the following generators of the algebra:

rx = R2x (8.1)

and

wx,x+µ = RxWx,x+µRx+µ , (8.2)

which, in terms of gauge-dependent fields, reads:

wx,x+µ = φxeigaAx,x+µφx+µ . (8.3)

Of course,

wx+µ,x = wx,x+µ . (8.4)

It is easy to obtain the following, complete set of commutation relations between thesegenerators:

[rx, rx+0

]=

2ihτa3

(wx,x+0 + wx,x+0

)

[rx, wx,x+0

]=

[rx, wx,x+0

]=

2ihτa3

rx[rx+0, wx,x+0

]=

[rx+0, wx,x+0

]= −2ihτ

a3rx+0

33

[rx+0, wx,x+k

]= −2ihτ

a3wx,x+0 r

−1xwx,x+k

[rx, wx+0,x+0+k

]=

2ihτa3

wx,x+0 r−1x+0 wx+0,x+0+k

[wx,x+0, wx,x+k

]= −2ihτ

a3wx,x+k

[wx,x+0, wx,x+k

]= 0

[wx,x+0, wx+0,x+0+k

]= 0

[wx,x+0, wx+0,x+0+k

]= −2ihτ

a3wx+0,x+0+k

[wx+k,x+k+l, wx+0,x+0+k

]=

2ihτa3

wx+k,x+k+l wx+0,x+0+k r−1x+0+k

wx+k,x+0+k r−1x+k

(8.5)

wx+0,x+0+k wx,x+k = eihg2 τa wx,x+k wx+0,x+0+k

wx+0,x+0+k wx+k,x = eihg2 τa wx+k,x

(wx,x+0 −

2ihτa3

)r−1xwx,x+0 r

−1x+0 ×

× wx+0,x+0+k r−1x+0+k

wx+k,x+0+k r−1x+k

(wx+k,x+0+k −

2ihτa3

)

(8.6)

8.2 Second order field equations

Field equations may also be rewritten as second order difference equations for the inde-pendent set of configuration variables rx and wx,x+µ described above. At every lattice sitewe will have an equation analogous to the Klein-Gordon-like equation (2.7) for the “mat-ter” field r and at every lattice link an equation analogous to the Maxwell equation (2.7)for the “electromagnetic” field.

First notice, that the fourth of equations (6.8) is already in a form very similar tothe classical expression. We will express the momentum K in terms of the configurationvariables using the definition (6.10) of Wx,x+0. Indeed, we have:

wx,x+0 = R2x + τ

(K+x

+ iM+x

+ ih

a3

). (8.7)

Taking the real and the imaginary parts we get:

K+x

=1τ

[wx,x+0 + wx,x+0

2− rx

](8.8)

34

and

M+x

=1τ

wx,x+0 − wx,x+0

2i− h

a3. (8.9)

In the same way we obtain

K−x

= −1τ

[wx,x−0 + wx,x−0

2− rx

](8.10)

and

M−x

= −1τ

wx,x−0 − wx,x−0

2i− h

a3. (8.11)

Substituting the above expressions for K to (6.8), we have

−V′ (rx) · r12x

+∑

µ

1a(µ)2

[wx,x+µ + wx,x+µ

2− rx

]= 0 , (8.12)

which is the lattice version of the first equation of (2.7) for the quantum observables.To obtain the analog of the second of equations (2.7), let us choose a space-like lattice

link (x, x+ k). The last equation (6.8) is completely analogous to the classical expression,so we merely rewrite it in another form:

1a

∑

l

Bx;k,l −1τD+x,x+k

+1τD−x,x+k

= −g 1a

wx,x+k − wx,x+k

2i. (8.13)

To do the same for time-like links, we substitute (8.9) to the definition (5.9) of M :1a

∑

k

D+x,x+k

= g1τ

wx,x+0 − wx,x+0

2i− g h

a3=

= g1τr− 1

2x

wx,x+0 − wx,x+0

2ir

12x.

(8.14)

Summarizing, we may write the expressions (8.13) and (8.14) in one formula:∑

ν

gνν

a(ν)fx;µ,ν = −g 1

a(µ)r− 1

2x

wx,x+µ − wx,x+µ

2ir

12x, (8.15)

analogous to the second equation of (2.7).Finally, the value of the field fx;µ;ν may be expressed in terms of r and w (up

to 2πnga(µ)a(ν)). For both µ and ν being space-like, this is given simply by the constraint (3.12).

If one of them is time-like, this is given by the second of equations (6.14):

eihg2 τ

2awx+k,xr−1xwx,x+0r

−1x+0wx+0,x+0+kr

−1x+0+k

wx+0+k,x+kr−1x+k

= e−igaτD+

x,x+k , (8.16)

or, equivalently:

e−ihg2 τ

2awx,x+0r−1x+0wx+0,x+0+kr

−1x+0+k

wx+0+k,x+kr−1x+k

wx+k,xr−1x

= e−igaτD+

x,x+k . (8.17)

The above equations are equivalent to the definition of fµν as the curl of vµ, up to a vortexcarried by the gradient of argφ (see equation (2.6)).

35

9 Conclusions and perspectives

Each step of the kinetic evolution is given by the unitary operator eiτhHkin , where

Hkin :=a3

2

∑x

R−1x

K2

x +(

1ga3

Qx

)2R−1

x+a3

2

∑

x,k

D2x,x+k . (9.1)

Similarly, each step of the potential evolution is given by the unitary operator eiτhHpot ,

where

Hpot := a3∑x

V(R2x

)+

+a

2

∑

x,k

R2

x +R2x+k − 2RxRx+k

Wx,x+k

+W−1x,x+k

2

+

a3

2

∑

x;k,l

B2x;k,l .

(9.2)

Passing to the limit τ → 0 we obtain a finite dimensional quantum mechanical systemwith the degrees of freedom described by the algebra OΣ. Due to the Trotter formula,the dynamics of the system is defined by the total Hamiltonian

H = Hkin +Hpot . (9.3)

The Hamiltonian is manifestly positive. For a finite lattice one should first find its lowest-energy state, which plays the role of the physical vacuum. Because of the high non-linearity of the theory, only numerical analysis will probably be possible. The physicalvacuum has nothing to do with the perturbative vacuum, which may be defined as a tensorproduct of the free-electromagnetic-vacuum and the free-matter-vacuum. For the sake ofconvenience one can subtract the vacuum energy from the Hamiltonian.

Then, one has to renormalize the mass and possibly other physical parameters inthe original potential V, i.e. to relate them with their physical values. To calculate thephysical mass of the particles in our theory we can not use the one-particle states as in theperturbative approach. Indeed, there is no “creation operator” in our theory, because ofthe super-selection rules, which are automatically fulfilled. Instead, the operator Wx,x+kcreates a particle carrying the charge −e at x and an antiparticle carrying the charge +eat x + k. Therefore, one has to proceed as follows. Fix a pair of sites (x, y) ∈ Σ. In thesubspaces of all quantum states, such that Qx = +e, Qy = −e, and Q = 0 elsewhere, wefind the state of the lowest energy E(x, y). The maximal value of the function E(x, y)with respect to x and y will be identified with twice the physical mass of the particles wewant to describe. This value has to be fitted by the parameters of V.

Having chosen appropriate parameters of the theory, one may further investigate theproperties of the Hamiltonian, possibly by numerical methods. In particular, dynami-cal problems may be solved discretizing again the time and using the unitary evolutiondeveloped in this paper.

36

Finally, one has to check, whether or not the physical properties of the above systemdepend considerably on the volume and the spacing a of the lattice used.

A similar program for Quantum Electrodynamics with spinorial matter is now underinvestigation and will be presented in the next paper.

Acknowledgments

The authors are very much indebted to Prof. I. Bia lynicki-Birula, Dr G. Rudolph and DrS. Zakrzewski for many fruitful discussions.

The authors are also very much indebted for the financial support, which they gotfrom the European Community (HCM Contract No. CIPA–3510–CT92–3006) and fromthe Polish National Committee for Scientific Research (Grant No. 2 P302 189 07).

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