A Lattice Approximation of Dirac Equation

Transcription

1 A Lattice Approximation of Dirac Equation Jerzy Kijowsi and Artur Thielmann Institute for Theoretical Physics Polish Academy of Sciences Al. Lotniów 3/46, Warsaw, Poland ABSTRACT A different point of view on discretisation of classical theory of Dirac equation is given. Canonical structure of the model is given, Cauchy problem is formulated and solved, fermion doubling is discussed and a solution via time conserved constraints is proposed. 1. Introduction The aim of the present paper is to construct a lattice approximation of the Dirac equation, which preserves as much as possible of the specific properties of the continuous Dirac theory, such as canonical structure, the structure of the space of initial data, etc. The formulation is meant as a starting point for quantization. Various versions of discrete approximations of spinor field theories have already been proposed (see e.g., [1-4]). The main disease of these approximations lies in doubling of the number of solutions with respect to the original Dirac equation. This means that the lattice version of the Dirac equation becomes rather a second order than a first order equation. In general there is no satisfactory lattice description of first order partial differential equations. Our idea is to interprete Dirac theory as a combination of two ingredients: 1) a second order dynamical equation and ) a constraint imposed on initial data (see section ). We show that a similar idea can be realized on the level of the lattice approximation of the theory. This way we obtain the theory without any fermion doubling. The theory is constructed on a four-dimensional, Minowsi space hypercubic lattice Λ with sites at the points R 4 x = δ n, where δ denotes the lattice constant and n = (n 0, n 1, n, n 3 ) Z 4 is a point in R 4 with integer coordinates. To represent momenta canonically conjugate to field variables we will also need the geometrically dual lattice Λ, i.e., the lattice whose sites are precisely the centers of the four-dimensional hypercubes of Λ. Finally, the complete version of the theory will be given on the full composed lattice Λ Λ. Sites of Λ will be denoted by x,..., lins (x, x ˆµ),... or (x; ˆµ),..., where ˆµ is a vector of length δ and direction of the oriented µ-th axis.

2 . Continuous theory.1. Lagrangian formulation The classical theory of a complex bispinor field ψ(x) can be derived from the lagrangian L = i ( ψγ µ µ ψ µ ψγ µ ψ ) mψψ, (.1) which implies the following second-type constraint p µ = δl δ µ ψ = i γµ ψ (.) relating the canonical momentum with the field variable. The field equation derived from the lagrangian (.1) is the Dirac equation δl δl µ = i ( δψ δ µ ψ γµ µ ψ mψ µ i γ µ ψ ) = iγ µ µ ψ mψ = 0. (.3).. Hamiltonian formulation Now we pass to the hamiltonian formulation of the theory. It is based on the hamiltonian obtained from the above lagrangian via the standard Lagrange transformation ( ) H = p 0 0 ψ 0 ψp 0 L = i ψγ ψ ψγ ψ mψψ. (.4) The hamiltonian is defined on the phase space P consisting of initial data (p 0, ψ) which fulfill the constraint equation (.). Following [8] we parameterize the phase space by the real q and the imaginary p parts of the bispinor ψ (in the sense of the charge conjugation). For the sae of simplicity we use the Majorana representation, where the charge conjugation coincides with the complex conjugation. This means that both q and p are four-dimensional real bispinors. Substituting ψ = q ip to the hamiltonian (.4) we obtain H = p T γ 0 γ q imp T γ 0 q. (.5) The canonical structure of the phase space of the theory is given by the standard symplectic form ω = dp 0 dψ dψ dp 0 which reduces on the subspace defined by the constraint (.) to ω = idψγ 0 dψ. Then in terms of the unconstrained variables q and p one gets ω = dp T dq.

3 3 This means that p is now the momentum canonically conjugate to q. Therefore, the field dynamics is governed by the Hamilton equations q = 1 δh δp T ṗ = 1 (.6) δh δq T. Using the specific form (.5) of the hamiltonian the reader may easily chec that these are precisely the real and the imaginary parts of the Dirac equation iγ µ µ q = mq iγ µ µ p = mp. (.7).3. Complexification of the phase space Consider now the complexification P of the above phase space and define on it the hamiltonian by the formula H = pγ q impq qγ p imqp, (.8) where q and p are no longer real. Obviously, the hamiltonian H coincides with H on the subspace P P. Define also the canonical structure in P by the following symplectic form Ω = dpγ 0 dq dqγ 0 dp, which on P coincides obviously with ω. The hamiltonian field equations of this theory tae the form q = δh δp ṗ = δh (.9) δq coinciding practically with equations (.7), except that now both q and p are complex. The theory is thus the doubling of the original Dirac theory. Limiting ourselves to the subspace of real data q = q p = p (.10) at one instant of time, we will remain forever in this subspace, since it is invariant with respect to the dynamics. This way we reproduce the original theory as a sub-theory corresponding to the invariant subspace P in the phase space P of the doubled Dirac theory. As we show later, similar construction can also be performed in the lattice version of the theory.

4 4 3. Primary lattice approximation The action of the Dirac theory defined by (.1) we approximate on Λ by the following expression A = { [ ] [ ] i ψ(xˆµ)ψ(x) δ4 γ µ ψ(xˆµ) ψ(x) δ (x,ˆµ) [ ] ψ(xˆµ) ψ(x) δ [ ] } γ µ ψ(xˆµ)ψ(x) mδ 4 x ψ (x) ψ (x). (3.1) In the above formula the derivative related to a lattice lin is approximated by the simple difference term and the value of the field on the lin by the arithmetic mean. We could approximate ψ by the value at one of the lin s endpoints, but this would differ from our formula only by a rearrangement of terms in A (in continuous theory this corresponds to adding a boundary integral to the action). Such a change does not influence the field equations. The field equations resulting from the action (3.1) are 0 = A ψ(x) = iδ3 or rewritten in a different form ψ (x ˆµ) = ψ (x ˆµ) γ µ [ψ (x ˆµ) ψ (x ˆµ)] mδ 4 ψ (x) (3.) µ γ 0 γ [ ψ ( x ˆ ) ( ψ x ˆ )] iδ mγ 0 ψ (x). (3.3) In the limit δ 0 they tend to the Dirac equation. Unfortunately they combine the variables ψ at three consecutive space-lie hypersurfaces, that of x ˆ0, that of x itself and that of x ˆ0 (ˆ0 is a vector of the length δ and the direction of the oriented 0-th axis). As a dynamical equation (3.3) is therefore of the second order. To find a unique solution of this equation we have to specify initial values of ψ at two consecutive instants of time or, in other words, the value of the function and of its time derivative. There are therefore twice as many initial conditions (and consequently twice as many solutions) as we would expect for Dirac equation. For this reason the primary lattice description is not satisfactory and we will loo for a more detailed, canonical structure of the lattice model.

5 5 4. Dual lattice theory 4.1. Construction of the action According to the symplectic philosophy of field theories (see e.g., [5-7]) we describe field dynamics as a superposition of local dynamics related to each spacetime cell separately. Within a single lattice cell the dynamics is expressed in terms of the local configurations and momenta. The composition of the local dynamics corresponding to neighboring cells follows from the matching condition on their common wall. The condition states that both the configurations and momenta on the wall are equal. To construct the lattice version of the phase space P we have to represent infinitely many degrees of freedom of the field in a spacetime cell by a finite number of configurations and momenta. Whereas the configurations were described on the primary lattice, the momenta need the use of the dual lattice. This is due to the fact that geometrically the momenta are vector densities or differential three-forms in spacetime i.e., objects which have to be integrated over three-dimensional domains (e.g., walls of lattice cells). It is natural therefore to represent configurations with their sample values on primary lattice elements and momenta as averaged over the corresponding dual lattice elements (see [5]). We add therefore the configurations ψ(x ˆµ ) at the centers of lins x ˆµ to the description of field configuration at the primary lattice sites ψ(x). The points x ˆµ are also centers of the three-dimensional walls of the dual lattice cells. To find momenta conjugate to both types of variables (configurations at the primary lattice sites and at the centers of lins), we want to rewrite the action (3.1) of the theory in terms of these variables. We would lie to have the action in the form A = δ 4 L x, (4.1) x with local lagrangians L x depending on configurations contained in the lattice cell K x dual to the site x only, i.e. on ψ(x) and on ψ(x ˆµ ) at all centers of the lins outgoing from x. To eep the dynamics on the primary lattice unchanged, we want our new action to reduce to the old one when the new variables ψ(x ˆµ ) are eliminated. By the elimination we mean the variation with respect to ψ(x ˆµ ) and the substitution of the stationary value to the action. The equation obtained by the above variation relates the value of ψ at the center of a lin to the values at its endpoints and may therefore be called interpolation equation. The reduction requirement guarantees that variation of the new action with respect to the primary lattice sites configurations will reproduce the previous approximation of the Dirac equation. Assuming that the action is quadratic with respect to the variables ψ(x) and ψ(x ˆµ ) connecting only nearest neighbors and assuming some symmetry conditions we have the solution unique up to a change of sign of the imaginary unit i.

6 6 We start from a general expression for a quadratic form of the variables ψ(x) and ψ(x ˆµ ) connecting only nearest neighbors L x = { a ψ (x) γ µ ψ x ˆµ b ψ (x) γ µ ψ x ˆµ µ cψ x ˆµ γ µ ψ (x) dψ x ˆµ γ µ ψ (x) } fψ (x) γ µ ψ (x) eψ x ˆµ γ µ ψ x ˆµ eψ x ˆµ γ µ ψ x ˆµ mψ (x) ψ (x). (4.) We decided the last two terms under the summation sign to have the same coefficient e, since they simply add up in the sum (4.1) of neighboring cells actions. First, we want the action to be real, so we have a = c b = d e = e f = f. Variation of (4.) with respect to ψ(x ˆµ ) gives the interpolation equation eψ x ˆµ = cψ (x) dψ (x ˆµ). Eliminating all ψ(x ˆµ ) from the action we get ea = δ { 4 ad ψ (x) γ µ ψ (x ˆµ) bc ψ (x ˆµ) γ µ ψ (x) x µ } [ac bd ef] ψ (x) γ µ ψ (x) emδ 4 x ψ (x) ψ (x). (4.3) Comparing this to the primary lattice formula (3.1) we get the following system of equations for coefficients a a b b = ef ab = i δ e a b = i δ e, (4.4a) (4.4b) (4.4c) where the third equation is the complex conjugate of the second one (since e is real). Because the terms in the action containing a and b differ only by lin orientation, we assume that these coefficients have the same modulus. Decompose a and b

7 7 into the moduli and phase factors a = r exp(iϕ) and b = r exp(iχ), and substitute to (4.4b): r exp (iϕ) r exp ( iχ) = i δ e. (4.5) Taing moduli of both sides of (4.5) we obtain e = δr. Comparing the phase 1δ factors instead we get b = sgne ia. From (4.4a) we have now f = sgne. We substitute all the coefficients to the lagrangian (4.) and calculate the momentum. To obtain the momentum with a proper continuum limit we have to compare the variation of the action related to a single cell δ 4 L x with respect to the boundary configuration and the momentum integrated over the corresponding wall ( δ 4 ) L x =: δ 3 p x ˆµ ψ x ˆµ. This way p x ˆµ L x := δ ψ ( x ˆµ ) = δr exp ( iϕ) γ µ ψ (x) sgne δ r γ µ ψ x ˆµ Comparing the result with the continuum expression (.) we see that r has to behave lie 1 δ when δ 0, in order to provide the correct continuum limit of our theory. The simplest choice is δr = R. Then in the limit we obtain p = [ R exp ( iϕ) sgne R ] γ µ ψ and therefore the expression in the square bracets has to be equal to i. The real part gives R = sgne cos ϕ whereas the imaginary part R sin ϕ = 1. Performing the same operations with the opposite wall momentum p(x ˆµ ) we get R = sin ϕ and sgne R cos ϕ = 1. Since R is positive we have the unique solution for R = 1 and exp(iϕ) = 1i. The only choice which is still left is the sign of e. It is easy to chec that a change of the sign corresponds to the change of i to i everywhere. We choose e to be negative. Finally we have a = d = 1 i δ b = c = 1 i δ e = 1 δ f = 1 δ..

8 8 L x = i δ This way the dual lattice action becomes A = δ 4 x L x with { (1 i) ψ (x) γ µ ψ µ x ˆµ (1 i) ψ (x) γ µ ψ x ˆµ (1 i) ψ x ˆµ γ µ ψ (x) (1 i) ψ x ˆµ γ µ ψ (x) } iψ x ˆµ γ µ ψ x ˆµ iψ (x) γ µ ψ (x) iψ x ˆµ γ µ ψ x ˆµ mψ (x) ψ (x). (4.6) 4.. Lagrangian formulation Variation of the constructed action over ψ(x ˆµ ) gives the interpolation equation δa 0 = ( δψ x ˆµ ) = iδ3 [ ] γµ (1 i) ψ (x) (1 i) ψ (x ˆµ) iψ x ˆµ = [ 1 i = δ 3 γ µ ψ (x) 1 i ) (x ] ψ (x ˆµ) ψ ˆµ, (4.7) which reproduces exactly the original action (3.1) when substituted to (4.6). Now, the momentum canonically conjugate to ψ(x ˆµ ) corresponds to the threedimensional wall of an elementary lattice cell and is given by variation of the single-cell lagrangian p x ˆµ L = δ ( x ) [ ] = i γµ (1 i) ψ (x) iψ x ˆµ = ψ x ˆµ = i γµ ψ (x) i δ ψ x ˆµ δ ψ(x). (4.8) It is easy to see that the continuum limit of (4.8) provides the standard second-type constraint (.), because the additional term in the square bracets corresponds to the derivative of ψ multiplied by δ so it tends to zero in the continuum limit. Comparing the two expressions (4.7) and (4.8) one can notice, that the interpolation equation is equivalent to the requirement that the momenta corresponding to the neighboring cells are equal, i.e. p x ˆµ p y ˆµ = 0 (4.9) for y = x ˆµ.

9 9 The variation of (4.6) with respect to ψ(x) gives us the field equation [ 1 i ) γ (x µ δ ψ ˆµ 1 i ) (x δ ψ ˆµ 1 ] δ ψ (x) mψ (x) = 0. (4.10) µ The above expression tends to the ordinary Dirac equation (.3) when δ 0. To prove this, let us eliminate the configurations at the centers of lins using the interpolation formula (4.7) [ ] 1 i µ γ µ { 1 i δ 1 i δ [ 1 i ψ (x) 1 i ψ (x ˆµ) ψ (x ˆµ) 1 i ψ (x) This way we obtain the primary lattice expression (3.). ] 1δ } ψ (x) mψ (x) = Hamiltonian formulation We will describe now the time evolution of the Cauchy data. Our dynamic equation (4.10) is a second-order difference equation, so the Cauchy data consist of configurations and momenta on the initial hyperplane composed from the fully space-lie three-dimensional walls of the elementary cells of the lattice Λ. Using equations (4.7) (4.10) we compute Cauchy data on the later hyperplane Σ 1 through the data on the earlier hyperplane Σ 0. a ˆ a = x ˆ0 a ˆ Σ 1 x ˆ x x ˆ t Fig. 1 b ˆ b = x ˆ0 b ˆ Σ 0 Denoting the points as on Fig. 1 and denoting π := γ 0 p we have ψ a = imδγ 0 ψ b i ( 1 mδγ 0) π b 1 γ 0 γ ψ bˆ π bˆ ψ b ˆ π b ˆ (4.11a) π a = i ( 1 mδγ 0) ψ b imδγ 0 π b 1 γ 0 γ ψ bˆ π bˆ ψ b ˆ π b ˆ. (4.11b)

10 10 We introduce new variables ξ x ˆµ = ψ x ˆµ η x ˆµ = i ψ x ˆµ π ( i π ) x ˆµ (4.1) x ˆµ Let us notice, that vanishing of η is by definition equivalent to the constraint (.) in the continuous theory. To understand the meaning of the new parameters we express them in terms of the primary lattice variables using (4.7) and (4.8). We get ξ x ˆµ = ψ (x) ψ (x ˆµ) (4.13) η x ˆµ = ψ (x) ψ (x ˆµ). As we expected the second equation assures vanishing of η in the continuum limit. We can now rewrite Dirac equation (4.10) in the following form ξ a ξ b 1 γ 0 γ ξ aˆ ξ a ˆ ξ bˆ ξ b ˆ η a η b 1 γ 0 γ η aˆ η a ˆ η bˆ η b ˆ = = imδγ 0 (ξ a ξ b η a η b ) (4.14) and the interpolation condition as ξ a η a = ξ b η b. (4.15) It is easy to chec that combining the above two equations we get ξ a = ( 1 imδγ 0) ξ b imδγ 0 η b [ ] 1 γ 0 γ ξ bˆ η bˆ ξ b ˆ η b ˆ η a = ( 1 imδγ 0) η b imδγ 0 ξ b [ ] 1 γ 0 γ ξ bˆ η bˆ ξ b ˆ η b ˆ. (4.16) Observe that the same result can be obtained directly rewriting (4.11) or (3.3) in terms of ξ and η. In the limit when δ 0 the second equation implies vanishing of η, whereas the first one becomes Dirac equation for ξ. These are two equations for the complete complex bispinors.

11 Momentum picture It is convenient for our purposes to rewrite the lattice field dynamics (4.16) in terms of the Fourier transformations of the fields. In the present paper we assume that the lattice is infinite. In this case the space dual to the three-dimensional lattice (on which we defined the Cauchy data) is the three-dimensional torus Tδ 3 parameterized by three angles ] φ 1, φ, φ 3 π δ, π ]. δ Every function on the lattice can be written in the standard form of the Fourier transformation ξ b = d 3 φ e ib φ α (φ) (4.17a) η b = T 3 δ T 3 δ d 3 φ e ib φ β (φ), (4.17b) where b = (( n 0 ) 1 δ, n 1 δ, n δ, n 3 δ ) for a fixed n 0 and n Z, = 1,, 3. In the case of a finite lattice we would have 1 n N. The dual space is the three-dimensional cyclic lattice parameterized by the angles φ such that for every = 1,, 3 we have exp ( in ) φ = 1. The integrals over the torus coordinates would be substituted with finite sums. Otherwise all the formulae will remain unchanged. For the infinite lattice let us denote σ = γ sin (δ φ ). The evolution is given by the following operator (we omit here the obvious dependence of the variables α and β on the angles φ) ( α 1 iγ (t δ) = i 0 (σ mδ) iγ 0 ) (σ mδ) α β iγ 0 (σ mδ) 1 iγ 0 (t). (σ mδ) β Introduce the following operator Γ = iγ 0 (σ mδ). Its square is proportional to the unit matrix with the coefficient Γ = (mδ) σ < 0. The evolution operator simplifies now to α (t δ) = β ( 1 Γ Γ Γ 1 Γ ) ( α β ) (t).

12 1 5. Constraint equation As we have pointed out in the previous section, Dirac equation in the lattice version is a second-order difference equation. The Cauchy data for such a theory consist both of configurations and canonical momenta. Therefore, we have twice as many solutions as we would expect for a first-order equation (that we have in the standard continuum case). This manifests the fermion doubling problem in our model. The situation is similar as in the case of the complexified phase space we described in the paragraph.3., where the initial data were composed of two independent complex bispinor fields. Only solutions fulfilling the constraint equation (.10) were physically admissible. The constraint (.10) is conserved by the dynamics (.9). Once fulfilled by initial data, it will always be valid during the evolution. Our idea to remove the fermion doubling consists in reproducing this mechanism on the lattice level. We loo for a subspace P of the phase space P having the following two properties: 1) it is parameterized by a single complex bispinor field (corresponding to the two real fields q and p in the continuum case), ) it is conserved by the dynamics (4.16). Observe, that the dynamics (4.16) is given by real matrices, so the real and the imaginary parts of ξ and η do not interact during the evolution. Therefore a constraint ξ = ξ or a constraint η = η ξ = ξ η = η or their arbitrary linear combinations are conserved. Due to (4.13) this would correspond in the continuum limit to the constraint Imψ = 0 or Reψ = 0. Such a theory is not a complete continuous Dirac theory and therefore the idea to define P via constraint of this type has to be abandoned. Local constraints mixing ξ and η at one point are not conserved, so we have to loo for another possibility. Our idea is to mimic the second-type constraint (.) of the continuum theory. Assume that the constraint is given by a linear operator A: { } α P := β = Aα. β Assuming that the constraint P is conserved by the dynamics (4.1) we obtain the following condition on A α 1 Γ Γ α (t δ) = (t), Aα Γ 1 Γ Aα or equivalently A (1 Γ) A ΓA = Γ (1 Γ) A.

13 13 The simplest solution of this equation is obtained assuming AΓ = ΓA. In this case (5.1) implies 1Γ A = 1 Γ. Γ Rewriting the formula (5.) in terms of the original fields ξ and η we obtain the following, nonlocal constraint for the Cauchy data η (b 0 ) = i d 3 φ e i(b 0 b) φ 1 1 (mδ) σ γ 0 (σ mδ) ξ (b) b Σ 0 T 3 δ (mδ) σ with σ = sin (δ φ ). This constraint defines a linear subspace of the Cauchy data conserved by the time evolution. The solutions fulfilling (5.) on the initial hyperplane will remain in the same subspace forever, so the number of solutions of the lattice Dirac equation is reduced to the expected value, i.e. the doubling is canceled. To show that the formula (5.) provides the constraint (.) in the continuum limit, observe that the fraction in the integrand tends to 1 when δ 0. Since the term σ mδ vanishes as δ, the constraint assures vanishing of η. This is equivalent (see formula (4.1)) to the classical constraint (.). Finally, the theory based on the phase space P and the evolution (4.16) is the desired lattice version of the Dirac theory, without fermion doubling and with the proper continuum limit. REFERENCES 1. Wilson, K. G., Phys. Rev. D10, 8, 445 (1974).. Sussind, L., Phys. Rev. D16, 10, 3031 (1977). 3. Rabin, J. M., Nucl. Phys. B01, 315 (198). 4. Rossi, P., Wolff, U., Zwanziger, D., Phys. Rev. D30, 10, 33 (1984). 5. Kijowsi, J. and Tulczyjew, W. M., A Symplectic Framewor for Field Theories, Lecture Notes in Physics Vol. 107, Springer-Verlag, Berlin, Kijowsi, J. and Rudolph, G., Rep. Math. Phys. 1, 3, 309 (1985). 7. Kijowsi, J. and Rudolph, G., Lett. Math. Phys. 15, 119 (1988). 8. Jaubiec, A., Lett. Math. Phys. 9, 171 (1985).