Wigner - Weyl formalism and the propagator of Wilson fermions in the presence of varying external electromagnetic field
aa r X i v : . [ h e p - l a t ] N ov Wigner - Weyl formalism and the propagator of Wilson fermions in thepresence of varying external electromagnetic field
M.Suleymanov a , M.A. Zubkov a Physics Department, Ariel University, Ariel 40700, Israel
Abstract
We develop Wigner - Weyl formalism for the lattice models. For the definiteness we consider Wilson fermionsin the presence of U (1) gauge field. The given technique reduces calculation of the two point fermionic Greenfunction to solution of the Groenewold equation. It relates Wigner transformation of the Green functionwith the Weyl symbol Q W of Wilson Dirac operator. We derive the simple expression for Q W in the presenceof varying external U (1) gauge field. Next, we solve the Groenewold equation to all orders in powers of thederivatives of Q W . Thus the given technique allows to calculate the fermion propagator in the lattice modelwith Wilson fermions in the presence of arbitrary background electromagnetic field. The generalization ofthis method to the other lattice models is straightforward.
1. Introduction
The Wigner - Weyl formalism was originally proposed for the reformulation of ordinary quantum me-chanics. Later it was also adopted in some form both for the quantum field theory and for the condensedmatter physics theory. This formalism in its original form has been developed by H. Groenewold [1] and J.Moyal [2]. It is based on the notions of the Weyl symbol of operator and the Wigner distribution function.The authors of [1, 2] used the ideas developed earlier by H. Weyl [3] and E. Wigner [4]. The Wigner - Weylformalism in quantum mechanics is often referred to as the phase space formulation. It is defined in phasespace that is composed of both coordinates and momenta. At the same time the conventional formulationuses either coordinate space or momentum space representations. In the phase space formulation the quan-tum state is described by the Wigner distribution (instead of a wave function), while the operator productis replaced by the Moyal product of functions defined in phase space.The phase space formulation of quantum mechanics reduces the operator formulation in coordinate ormomentum space to the formulation that deals with the ordinary functions of coordinates and momenta[5, 6]. There are a lot of various applications of the phase space formulation of quantum mechanics (forthe review see, for example, [7, 8]). At the present moment there exist the alternative versions of thisformulation, in which the main notion - the distribution function - is defined differently [9, 10]. The mostpopular version mentioned above operates with the Wigner distribution function W ( x, p ) [4]. Among thealternative versions we mention those discussed in [11, 12, 13, 14, 15].In the one - dimensional case the quantum mechanical Wigner distribution function W ( x, p ) depends onthe coordinate x and on momentum p . Distribution W ( x, p ) determines the probability that the coordinate x belongs to the interval [ a, b ]: P [ a ≤ x ≤ b ] = 12 π Z ba Z ∞−∞ W ( x, p ) dp dx In ordinary quantum mechanics each observable is represented by operator. Weyl symbol A W ( x, p ) ofoperator ˆ A is the function in phase space such that the expectation value of the observable with respect to Corresponding author, e-mail: [email protected] on leave of absence from Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow, 117259,Russia Preprint submitted to Elsevier November 21, 2018 he given distribution W ( x, p ) is [2, 16] h ˆ A i = 12 π Z A W ( x, p ) W ( x, p ) dp dx. The Wigner function may be expressed through the ordinary wave function ψ ( x ) as follows W ( x, p ) = Z dy e − ipy ψ ∗ ( x + y/ ψ ( x − y/ i∂ t W ( x, p, t ) = H ( x, p ) ⋆ W ( x, p, t ) − W ( x, p, t ) ⋆ H ( x, p )Here the star (or Moyal) product of the two functions f and g is defined as f ⋆ g = f exp (cid:18) i ←− ∂ x −→ ∂ p − ←− ∂ p −→ ∂ x ) (cid:19) g The left arrow above the derivative means that it acts on f while the right arrow means that it acts on g . By H ( x, p ) the Weyl symbol of the Hamiltonian is denoted. In general case the definition of the Weyl symbolof operator ˆ A is: A W ( x, p ) = Z dy e − ipy h x + y/ | ˆ A | x − y/ i As it was mentioned above, the Wigner - Weyl formalism has also been extended, at least partially, tothe quantum field theory. Namely, one may consider the analogue of the Wigner distribution function W Γ ( x, p ) = Z dy e − ipy h ¯ ψ ( x + y/ ψ ( x − y/ i Here ψ is a certain fermionic operator while Γ is an appropriate matrix in spinor and flavor spaces. (Obviously,the similar construction may be introduced also for bosons.) In particular, in this way the analogue of theWigner distribution function has been defined in QCD (see, for example, [17, 18] and references therein).Besides, the similar notion of the Wigner distribution has been implemented in the framework of the quantumkinetic theory based on the field theory [20, 21]. It was also used in certain noncommutative field theories[22, 23]. Applications of the Wigner function formalism to cosmology were discussed in [24]. Anotherapplications of this formalism may be found, for example, in [26], see also [25].Recently the Wigner - Weyl formalism has been adopted to the investigation of anomalous transport inthe quantum field theory [27, 28, 29, 30, 31, 32]. The main advantage of this latter approach is that theresponse of the nondissipative components of both vector and axial currents to the external field strengthare expressed through the topological invariants in momentum space. These quantities are not changed ifwe deform the system smoothly. Thus we are able to calculate the corresponding conductivities within thesimple noninteracting systems connected continuously with the more complicated interacting ones. Andthis gives the values of the conductivities for the given complicated systems. In particular, in this waythe absence of the equilibrium chiral magnetic effect [33] in the field theory has been proved [31] while theanomalous quantum Hall effect has been described rigorously both for the Weyl semimetals and for certain2 D and 3 D topological insulators [32]. Besides, the chiral separation effect (proposed in [34]) was rigorouslyre - derived [29] and the alternative derivation of chiral anomaly in lattice regularized quantum field theoryhas been given [27]. The fermion zero modes on vortices in the color superconductor phase of QCD havebeen considered using this technique in [30]. In addition, the scale magnetic effect proposed in [35] has beenconsidered in the framework of this formalism [28].It is worth mentioning that previously the momentum space topological invariants expressed in termsof the Green functions were used mainly in the condensed matter physics theory (for the review see [36,37, 38, 39, 40]). The momentum space topological invariants protect gapless fermions on the boundaries oftopological insulators [41, 42] and protect the bulk gapless fermions in Dirac and Weyl semi - metals [44, 43].The gapless fermions associated with topological defects and textures existing in the fermionic superfluids2re also described by momentum space topology [45]. In the context of relativistic quantum field theorymomentum space topology has been discussed, for example, in [46, 47, 48, 49, 50, 44, 51, 52, 53]. In [54] thelattice regularization of QFT with Wilson fermions was discussed from the point of view of the momentumspace topology. Appearance of the massless fermions at the intermediate values of bare mass parameter isrelated to the jump of the introduced momentum space topological invariant.The version of the Wigner - Weyl formalism of [27, 29, 31, 32] deals with the lattice regularized quantumfield theory. Here the two - point fermion Green function G ( p , q ) in momentum space M ( p , q ∈ M ) hasbeen considered. It obeys equation ˆ Q ( p − A ( i∂ p ))) G ( p , q ) = δ ( p − q )where ˆ Q ( p ) is the lattice Dirac operator written in momentum space while A ( x ) is external electromagneticfield. Its appearance in momentum representation is given by the pseudo - differential operator A ( i∂ p )realizing the so - called Peierls substitution. Notice, that in this formalism the imaginary time of theEuclidian formulation of the quantum field theory gives one of the components of the (discrete) coordinatespace while the corresponding momentum is the fourth component of momentum vector that belongs to M .Wigner transformation of the two point Green function plays the role similar to the Wigner distributionin quantum mechanical formulation of [1] and [2]. It may be defined in continuous momentum space for thelattice model in the way similar to the Wigner transformation of continuous coordinate space: G W ( x , p ) ≡ Z dqe i xq G ( p + q / , p − q /
2) (1)It obeys the Groenewold equation G W ( x , p ) ⋆ Q W ( x , p ) = 1 (2)with the same star operation as above extended to the D - dimensional vectors of coordinates x and mo-mentum p , while Q W is the Weyl symbol of operator ˆ Q ( p − A ( i∂ p )). (Compare Eq. (2) with Eq. (4.38) of[1] and Eq. (7.10) of [2], where the Weyl symbol of the commutator of two operators is represented.)In the present paper we proceed with the development of the approach of [27, 29, 31, 32]. For thedefiniteness we restrict ourselves with the lattice model with Wilson fermions in the presence of arbitraryexternal electromagnetic field. The obtained results may easily be extended to the other lattice models aswell as to various tight - binding models of solid state physics. First of all we derive the precise expressionfor the Weyl symbol of the Wilson Dirac operator. Next, we present the complete iterative solution ofthe Groenewold equation Eq. (2), which allows to calculate the fermion propagator in the background ofarbitrary external electromagnetic field.The paper is organized as follows. We start in Sect. 2 from the pedagogical introduction to the con-ventional notions of Wigner transformation, Weyl symbols of operators and Moyal product. In Sect. 3 weproceed with the summary of the momentum space formulation of [27, 29, 31, 32]. In Sect. 4 we derivethe Peierls substitution, that allows to calculate easily the Wilson Dirac operator in momentum space inthe presence of arbitrary electromagnetic field. In Sect. 5 we present our results on the Weyl symbol ofWilson Dirac operator. In Sect. 6 we propose the procedure of the iterative solution of the Groenewoldequation, which allows to calculate completely the propagator of Wilson fermions in the presence of externalelectromagnetic field. In Sect 7 we end with the conclusions.
2. Wigner transformation in continuous coordinate space and continuous momentum space
In this section we briefly review the technique of Wigner transformation applied to quantum mechanicsdefined in infinite continuous coordinate space.
We start from the definition of the average of operator ˆ A with respect to quantum state Ψ h Ψ | ˆ A | Ψ i = Z dxdy h Ψ | x i h x | ˆ A | y i h y | Ψ i (3)3ere by x, y we denote the continuous coordinates. For simplicity we consider the case of one - dimensionalspace R . The generalization of our expressions to the case of D - dimensional space R D is straightforward.Let us change the coordinates: x = u + v/ y = u − v/ dxdy = ∂ ( x, y ) ∂ ( u, v ) dudv = (cid:12)(cid:12)(cid:12)(cid:12) ∂x∂u ∂y∂u∂x∂v ∂y∂v (cid:12)(cid:12)(cid:12)(cid:12) dudv = − dudv (5)This gives h Ψ | ˆ A | Ψ i = − Z dxdy h x + y/ | ˆ A | x − y/ i h Ψ | x + y/ i h x − y/ | Ψ i = − Z dxdydz h x + y/ | ˆ A | x − y/ i δ ( z − y ) h Ψ | x + z/ i h x − z/ | Ψ i = − Z dxdydzdp h x + y/ | ˆ A | x − y/ i e − ip ( z − y ) π h x − z/ | Ψ i h Ψ | x + z/ i = Z dxdp π dye ipy h x + y/ | ˆ A | x − y/ i dze ipz h x + z/ | Ψ i h Ψ | x − z/ i (6)The Weyl symbol of operator ˆ A is defined as follows A W ( x, p ) ≡ Z dye − ipy h x + y/ | ˆ A | x − y/ i (7)or, in terms of the matrix elements in momentum space: A W ( x, p ) ≡ Z dqe iqx h p + q/ | ˆ A | p − q/ i (8)Here h x | p i = 1 √ π e ipx Wigner function for the state Ψ is defined as the Weyl symbol of the corresponding density operator: W ( x, p ) = ρ W ( x, p ) = Z dye − ipy h x + y/ | Ψ i h Ψ | x − y/ i (9)In momentum space we have: W ( x, p ) = ρ W ( x, p ) = Z dqe iqx h p + q/ | Ψ i h Ψ | p − q/ i (10)Hence, h Ψ | ˆ A | Ψ i = Z dxdp π A W ( x, p ) W ( x, p ) (11)Wigner function for the mixed state with the density matrixˆ ρ = X i | Ψ i i ρ i h Ψ i | may be defined as W ( x, p ) = ρ W ( x, p ) = Z dqe iqx X i h p + q/ | Ψ i i ρ i h Ψ i | p − q/ i (12)The vacuum average of operator ˆ A with respect to this mixed state may be written in the similar way h ˆ A i = Z dxdp π A W ( x, p ) W ( x, p ) (13)4 .2. Moyal product The Weyl symbol of the product of two operators is given by( AB ) W ( x, p ) = Z dye − ipy h x + y/ | ˆ A ˆ B | x − y/ i = Z dydze − ipy h x + y/ | ˆ A | z i h z | ˆ B | x − y/ i (14)The transformation of variables y = u + v z = x − u/ v/ dydz = ∂ ( y, z ) ∂ ( u, v ) dudv = (cid:12)(cid:12)(cid:12)(cid:12) ∂y∂u ∂z∂u∂y∂v ∂z∂v (cid:12)(cid:12)(cid:12)(cid:12) dudv = dudv (16)and ( AB ) W ( x, p ) = Z dudve − ip ( u + v ) D x + u v (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) x − u v E D x − u v (cid:12)(cid:12)(cid:12) ˆ B (cid:12)(cid:12)(cid:12) x − u − v E = (cid:20)Z due − ipu D x + u (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) x − u E(cid:21) e v ←− ∂ x − u −→ ∂ x (cid:20)Z dve − ipv D x + v (cid:12)(cid:12)(cid:12) ˆ B (cid:12)(cid:12)(cid:12) x − v E(cid:21) = (cid:20)Z due − ipu D x + u (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) x − u E(cid:21) e i (cid:16) ←− ∂ x −→ ∂ p −←− ∂ p −→ ∂ x (cid:17) (cid:20)Z dve − ipv D x + v (cid:12)(cid:12)(cid:12) ˆ B (cid:12)(cid:12)(cid:12) x − v E(cid:21) (17)
The Moyal product of the Weyl symbols of the two operators ˆ A and ˆ B is defined as follows A W ( x, p ) ⋆ B W ( x, p ) ≡ ( AB ) W ( x, p ) = A W ( x, p ) e i (cid:16) ←− ∂ x −→ ∂ p −←− ∂ p −→ ∂ x (cid:17) B W ( x, p ) (18)
3. Wigner transformation in discrete coordinate space and compact momentum space
In this section we discuss the generalization of the above mentioned constructions to the case, whencoordinate space is discrete and infinite, and, therefore, momentum space is continuous but compact.
Average of an operator with respect to state Ψ is defined as h Ψ | ˆ A | Ψ i = Z dpdq h Ψ | p i h p | ˆ A | q i h q | Ψ i (19)Here the integral is over the compact momentum space. The integration measure is defined in such a way,that Z dp = 2 π while h x n | p i = 1 √ π e ipx n for any lattice point x n . Again, we restrict ourselves by the one dimensional case. The generalization to themultidimensional systems is straightforward. Then h Ψ | ˆ A | Ψ i = − Z dpdq h p + q/ | ˆ A | p − q/ i h Ψ | p + q/ i h p − q/ | Ψ i = − Z dpdqdk h p + q/ | ˆ A | p − q/ i δ ( k − q ) h Ψ | p + k/ i h p − k/ | Ψ i = − X x n Z dpdqdk h p + q/ | ˆ A | p − q/ i e − ix n ( k − q ) π h p − k/ | Ψ i h Ψ | p + k/ i = X x n Z dp π dqe ix n q h p + q/ | ˆ A | p − q/ i dke ix n k h p + k/ | Ψ i h Ψ | p − k/ i (20)5he Weyl symbol of operator ˆ A is then defined as follows A W ( x n , p ) ≡ Z dqe ix n q h p + q/ | ˆ A | p − q/ i (21)And the Wigner function, which is the Weyl symbol of density operator, is W ( x n , p ) = Z dqe ix n q h p + q/ | Ψ i h Ψ | p − q/ i (22)hence h Ψ | ˆ A | Ψ i = X x n Z dp π A W ( x n , p ) W ( x n , p ) (23) Let us consider the Weyl symbol of the product of two operators ˆ A and ˆ B :( AB ) W ( x n , p ) = Z dqdke ix n ( q + k ) (cid:28) p + q k (cid:12)(cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12)(cid:12) p − q k (cid:29) (cid:28) p − q k (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (cid:12)(cid:12)(cid:12)(cid:12) p − q − k (cid:29) = (cid:20)Z dqe ix n q D p + q (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) p − q E(cid:21) e k ←− ∂ p − q −→ ∂ p (cid:20)Z dke ix n k (cid:28) p + k (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (cid:12)(cid:12)(cid:12)(cid:12) p − k (cid:29)(cid:21) = (cid:20)Z dqe ix n q D p + q (cid:12)(cid:12)(cid:12) ˆ A (cid:12)(cid:12)(cid:12) p − q E(cid:21) e i (cid:16) −←− ∂ p −→ ∂ xn + ←− ∂ xn −→ ∂ p (cid:17) (cid:20)Z dke ix n k (cid:28) p + k (cid:12)(cid:12)(cid:12)(cid:12) ˆ B (cid:12)(cid:12)(cid:12)(cid:12) p − k (cid:29)(cid:21) (24)Hence, the Moyal product may be defined as in the case of continuous space( AB ) W ( x n , p ) = A W ( x n , p ) e i (cid:16) ←− ∂ xn −→ ∂ p −←− ∂ p −→ ∂ xn (cid:17) B W ( x n , p ) (25)Notice, that Eqs. (21) and (22) define the Weyl symbol of operator ˆ A and the Wigner function as thefunctions of the continuous variable x n although the original coordinate space is discrete. This formaldefinition allows to use Eq. (25) with the derivative with respect to x n .As it was mentioned above, all considered expressions may be easily extended to the case of the D- dimensional system with the compact momentum space of arbitrary form. Then we assume that theintegration measure over momentum space is normalized in such a way, that Z dp = (2 π ) D while the states | x n i are defined in such a way that h x n | x m i = δ nm . With this normalization the definitionsof the Weyl symbol of operator and the Wigner function as well as the definition of Moyal product remainthe same. At the same time for the average of operator ˆ A with respect to the given quantum state we have h Ψ | ˆ A | Ψ i = X x n Z dp (2 π ) D A W ( x n , p ) W ( x n , p ) (26)For the mixed states given by the density matrix ˆ ρ = P i | Ψ i i ρ i h Ψ i | we have the Wigner function of theform W ( x, p ) = ρ W ( x, p ) = Z dqe iqx X i h p + q/ | Ψ i i ρ i h Ψ i | p − q/ i (27)Again, the vacuum average of operator ˆ A with respect to this mixed state may be written as h ˆ A i = Z dxdp (2 π ) D A W ( x, p ) W ( x, p ) (28)6 . Wilson fermions and Peierls substitution Let us consider the partition function for the lattice model with Wilson fermions written in the discretecoordinate space: Z = Z D ¯Ψ D Ψ exp − X r n , r m ¯Ψ( r m ) ( − i D r n , r m ) Ψ( r n ) ! (29)Here D x , y = − X i (cid:2) (1 + γ i ) δ x + e i , y + (1 − γ i ) δ x − e i , y (cid:3) U x , y + ( m (0) + 4) δ x , y (30) γ k are the Dirac gamma matrices, m (0) is the parameter that has the meaning of mass.Wilson fermions in momentum space correspond to the propagator of the form G ( p ) = ˆ Q − ( p )Here the operator ˆ Q has the form ˆ Q ( p ) = X k =1 , , , γ k g k ( p ) − im ( p ) (31)where g k ( p ) = sin( p k ) m ( p ) = m (0) + X ν =1 (1 − cos( p ν )) (32)The same system represents the cubic tight - binding model of certain solid state systems if momentumcomponent p is rescaled accordingly in order to represent the discretization of the imaginary time. Themodel may be extended in several ways to reproduce the tight - binding models for various real materials.Below we will show that coupling of this system to the electromagnetic field A ( x ) results in the so - calledPeierls substitution ˆ p → ˆ p − A ( i∂ p ). In momentum space the electromagnetic field appears as the pseudo -differential operator, in which the dependence of A on coordinate is substituted by the dependence on theoperator i∂ p . Let us consider the partition function for the lattice model with Wilson fermions in the presence of thegauge field written in discrete coordinate space: Z = Z D ¯Ψ D Ψ exp − X r n , r m ¯Ψ( r m ) ( − i D r n , r m ) Ψ( r n ) ! (33)Here D x , y = − X i (cid:2) (1 + γ i ) δ x + e i , y + (1 − γ i ) δ x − e i , y (cid:3) U x , y + ( m (0) + 4) δ x , y (34) γ k are the Dirac gamma matrices, m (0) is the parameter that has the meaning of mass, while U x , y = P e i R yx dξξξ A ( ξξξ ) (35)We restrict ourselves by the case of the U (1) gauge field A and then this parallel transporter is given by U x , y = e i R yx dξξξ A ( ξξξ ) (36)7 .3. Peierls substitution Let us denote D r n , r m = − X i (cid:2) (1 + γ i ) δ r n + e i , r m + (1 − γ i ) δ r n − e i , r m (cid:3) U r n , r m + ( m (0) + 4) δ x , y (37)and I = X r n , r m ¯Ψ( r m ) ( − i D r n , r m ) Ψ( r n ) (38)Also we define Ψ( r ) = Z M d D p |M| e i rp Ψ( p ) (39)where |M| = (2 π ) D is the volume of momentum space.The above expression for I contains the terms proportional to X r n , r m ¯Ψ( r m ) δ r n ± e i , r m e i R r m r n dξξξ A ( ξξξ ) Ψ( r n ) = X r n ¯Ψ( r n ± e i ) e i R r n ± e i r n dξξξ A ( ξξξ ) Ψ( r n ) = X r n (cid:20)Z M d D p |M| e − i ( r n ± e i ) p ¯Ψ( p ) (cid:21) e i R r n ± e i r n dξξξ A ( ξξξ ) (cid:20)Z M d D q |M| e i r n q Ψ( q ) (cid:21) = X r n Z M d D p |M| Z M d D q |M| ¯Ψ( p ) e i r n q e i R r n ± e i r n dξξξ A ( ξξξ ) e − i ( r n ± e i ) p Ψ( q ) = X r n Z M d D p |M| Z M d D q |M| ¯Ψ( p ) e i r n q exp (cid:20) − i Z ∓ e i dξ e i e iξ e i p A ( i∂ p ) e − iξ e i p (cid:21) e − i ( r n ± e i ) p Ψ( q ) = Z M d D p |M| ¯Ψ( p ) exp (cid:20) − i Z ∓ dξe iξp i A i ( i∂ p ) e − iξp i (cid:21) e ∓ i e i p Ψ( p ) (40)Here we used Eq. (116) from Appendix II with x = r n ± e i . Now let us apply Eq. (112) from the sameAppendix. This gives X r n , r m ¯Ψ( r m ) δ r n ± e i , r m e i R r m r n dξξξ A ( ξξξ ) Ψ( r n ) = Z M d D p |M| ¯Ψ( p ) e ∓ i e i ( p − A i ( i∂ p )) Ψ( p ) (41)We come to the partition function of the form Z = Z D ¯Ψ D Ψ exp (cid:18) − Z d D p |M| ¯Ψ( p ) Q ( p − A ( i∂ p ))Ψ( p ) (cid:19) (42)where Q is given by Eq. (31). One can see, that in momentum space the application of the external Abeliangauge field results in the Peierls substitution p → p − A ( i∂ p )8 . Weyl symbol for the Wilson fermions Below we will be interested in the Weyl symbols of the operators of the form of Q ( p − A ( i∂ p )), where A is given by Eq. (31). Let us start our consideration from the Weyl symbol for a general function of p ( f ( p )) W ( x , p ) = Z d q e i qx h p + q / | f ( p ) | p − q / i = Z d q e i qx f ( p − q / ) h p + q / | p − q / i = Z d q e i qx f ( p − q / ) δ ( q ) = f ( p ) (43)Weyl symbol for a general function of i∂ p is( g ( i∂ p )) W ( x , p ) = Z d q e i qx h p + q / | g ( i∂ p ) | p − q / i = Z d q e i qx h p + q / | g ( i∂ p ) | y i h y | p − q / i d y = Z d q e i qx h p + q / | g ( i∂ p ) | y i e i ( p − q / ) y (2 π ) / d y = Z d q e i qx g ( y ) h p + q / | y i e i ( p − q / ) y (2 π ) / d y = Z d q e i qx g ( y ) e − i ( p + q / ) y (2 π ) / e i ( p − q / ) y (2 π ) / d y = Z d q e i q ( x + y ) (2 π ) g ( y ) d y = g ( x ) (44)Here we used the fact that i ∂∂p | y i = y | y i . In these expressions for the definiteness we restricted ourselvesby the case of the three - dimensional momentum space. exp (cid:0) i (ˆ p k − A k e − ωωω∂ p ) (cid:1) Operator ˆ Q for the Wilson fermions is composed of the functions of a single momentum component.Moreover, those functions are sin and cos that are given by the sum of exponents. Therefore, in order tofind the Weyl symbol of ˆ Q it is enough to calculate the Weyl symbol of the exponent of the combination p k − A ( p , ..., p D ). We start our calculation from the consideration of F ( p , i∂ p ) = exp (cid:0) i ( p k − A k e − ωωω∂ p ) (cid:1) Let us calculate the commutator [ p k , A k e − ωωω∂ p ]: iA k e − ωωω∂ p ip k = i ( p k − ω k ) iA k e − ωωω∂ p (45)Therefore, [ i ˆ p k , iA k e − ωωω∂ p ] = iω k iAe − ωωω∂ p (46)Using the special case of BCH formula where [ X, Y ] = αY , we obtain: e X e Y = e X + α − e − α Y (47)This expression is given, for example, in Eq. (1.5) of [55]. The rather sophisticated derivation of this formulamay be found, for example, in [56]. We present our derivation of this formula in Appendix III. Changingvariables to Y ′ ≡ α − e − α Y = β − Y , where β − ≡ α − e − α , and [ X, Y ′ ] = αY ′ , we obtain: e X + Y ′ = e X e βY ′ (48)9n case of exp (cid:0) i (ˆ p k − A k e − ωωω∂ p ) (cid:1) with α = iω k we come toexp (cid:0) i (ˆ p k − A k e − ωωω∂ p ) (cid:1) = e i ˆ p k exp( − iβA k e − ωωω∂ p ) (49)This decomposition allows to calculate the Weyl symbol of F ( p , i∂ p ): h exp (cid:0) i (ˆ p k − A k e − ωωω∂ p ) (cid:1)i W = h e i ˆ p k exp( − iβA k e − ωωω∂ p ) i W = Z d q e i qx h p + q / | e i ˆ p k exp( − iβA k e − ωωω∂ p ) | p − q / i (50)Next, we use that i∂ p | y i = y | y i and i∂ p | p − q/ i = Z dyi∂ p | y i h y | p − q/ i = 1(2 π ) / Z dyy | y i e iy ( p − q/ = 1(2 π ) / Z dy | y i ( − i∂ p ) e iy ( p − q/ Therefore, e i ˆ p k exp( − iβA k e − ωωω∂ p ) | p − q / i = e i ˆ p k ∞ X n =0 ( − iβA k ) n n ! e − nωωω∂ p | p − q / i = e i ˆ p k ∞ X n =0 ( − iβA k ) n n ! | p − q / + nωωω i = ∞ X n =0 e i ( p k − q k / nω k ) ( − iβA k ) n n ! | p − q / + nωωω i (51)and come to h exp (cid:0) i (ˆ p k − A k e − ωωω∂ p ) (cid:1)i W = Z d q e i qx ∞ X n =0 e i ( p k − q k / nω k ) ( − iβA k ) n n ! h p + q / | p − q / + nωωω i = ∞ X n =0 e inωωω x e i ( p k + nω k / ( − iβA k ) n n ! = e ip k ∞ X n =0 e ni ( ωωω x + ω k / ( − iβA k ) n n ! =exp h i ( p k − βA k e i ( ωωω x + ω k / ) i = exp h i (cid:0) p k − (1 − e − iω k ) e iω k / iω k A k e iωωω x (cid:1)i =exp h i (cid:0) p k − sin( ω k / ω k / A k e iωωω x (cid:1)i (52) F = exp (cid:0) i (ˆ p µ − P NJ =1 A Jµ e k J i∂ p ) (cid:1) The next step is the consideration of the function F = exp (cid:0) i (ˆ p µ − P NJ =1 A Jµ e k J i∂ p ) (cid:1) , which appearsin the series expansion for arbitrary function (Fourier or Laplace series). As above, we will be using thefollowing relation: iA Jµ e k J i∂ p i ˆ p µ = i (ˆ p µ + ik Jµ ) iA Jµ e k J i∂ p (53)It gives [ i ˆ p µ , iA Jµ e k J i∂ p ] = k Jµ iA Jµ e k J i∂ p (54)Let us define the operator ˆ B Jµ ≡ A Jµ e k J i∂ p (55)It obeys the commutation relations [ ˆ B Jµ , ˆ B Iµ ] = 0 (56)and [ i ˆ p µ , i ˆ B Jµ ] = k Jµ i ˆ B Jµ (57)10herefore, we have [ i (ˆ p µ − B Jµ ) , iB Iµ ] = k Iµ iB Iµ (58)and h i (cid:16) ˆ p µ − X J ˆ B Jµ (cid:17) , i ˆ B Iµ i = k Iµ i ˆ B Iµ (59)As above we are able to use the special case of the BCH formula [55] for [ X, Y ] = αY , β = − e − α α e X + Y = e X e βY (60)We come to the following representation:exp (cid:16) i ˆ p µ − i N X J =1 ˆ B Jµ (cid:17) = exp (cid:16) i ˆ p µ − i N − X J =1 ˆ B Jµ − i ˆ B Nµ (cid:17) =exp (cid:16) i ˆ p µ − i N − X J =1 ˆ B Jµ (cid:17) exp (cid:16) − iβ Nµ ˆ B Nµ (cid:17) = e i ˆ p µ exp (cid:16) − i N X J =1 β Jµ ˆ B Jµ (cid:17) = e i ˆ p µ N Y J =1 e − iβ Jµ ˆ B Jµ (61)where β Jµ = 1 − e − k Jµ k Jµ (62)Hence, exp (cid:0) i (ˆ p µ − N X J =1 A Jµ e k J i∂ p ) (cid:1) = e i ˆ p µ N Y J =1 exp (cid:0) − iβ Jµ A Jµ e k J i∂ p (cid:1) = e i ˆ p µ N Y J =1 ∞ X n J =0 ( − iβ Jµ A Jµ ) n J n J ! e n J k J i∂ p = e i ˆ p µ ∞ X n =0 ( − iβ µ A µ ) n n ! ... ∞ X n N =0 ( − iβ Nµ A Nµ ) n N n N ! e ( n k + ... + n N k N ) i∂ p (63)For imaginary values of k we obtain h p ′′ | exp (cid:0) i (ˆ p µ − N X J =1 A Jµ e k J i∂ p ) (cid:1) | p ′ i = h p ′′ | e i ˆ p µ ∞ X n =0 ( − iβ µ A µ ) n n ! ... ∞ X n N =0 ( − iβ Nµ A Nµ ) n N n N ! e ( n k + ... + n N k N ) i∂ p | p ′ i = h p ′′ | e i ˆ p µ ∞ X n =0 ( − iβ µ A µ ) n n ! ... ∞ X n N =0 ( − iβ Nµ A Nµ ) n N n N ! | p ′ − i ( n k + ... + n N k N ) i = h p ′′ | ∞ X n =0 ( − iβ µ A µ ) n n ! ... ∞ X n N =0 ( − iβ Nµ A Nµ ) n N n N ! e ip ′ µ +( n k µ + ... + n N k Nµ ) | p ′ − i ( n k + ... + n N k N ) i = ∞ X n =0 ( − iβ µ A µ ) n n ! ... ∞ X n N =0 ( − iβ Nµ A Nµ ) n N n N ! e ip ′ µ +( n k µ + ... + n N k Nµ ) δ h ( p ′′ − p ′ ) + i ( n k + ... + n N k N ) i (64)11he last expression allows to calculate the Weyl symbol of F : h exp (cid:0) i (ˆ p µ − N X J =1 A Jµ e k J i∂ p ) (cid:1)i W = Z d q e i qx h p + q / | exp (cid:0) i (ˆ p µ − N X J =1 A Jµ e k J i∂ p ) (cid:1) | p − q / i = Z d q e i qx ∞ X n =0 ( − iβ µ A µ ) n n ! ... ∞ X n N =0 ( − iβ Nµ A Nµ ) n N n N ! e i ( p µ − q µ / n k µ + ... + n N k Nµ ) δ h q + i ( n k + ... + n N k N ) i = ∞ X n =0 ( − iβ µ A µ ) n n ! ... ∞ X n N =0 ( − iβ Nµ A Nµ ) n N n N ! e ( n k + ... + n N k N ) x e ip µ + ( n k µ + ... + n N k Nµ ) =exp h i (cid:0) p µ − N X J =1 β Jµ e k Jµ / A Jµ e k J x (cid:1)i (65)defining a ( k Jµ ) ≡ β Jµ e k Jµ / = 1 − e − k Jµ k Jµ e k Jµ / = sinh( k Jµ / k Jµ / h exp (cid:0) i (ˆ p µ − N X J =1 A Jµ e k J i∂ p ) (cid:1)i W = exp h i (cid:0) p µ − N X J =1 a ( k Jµ ) A Jµ e k J x (cid:1)i (67)This representation has been derived for the imaginary values of k . However, it may be extended to anycomplex values of k by analytical continuation. F = exp (cid:2) i (ˆ p µ − A µ ( i∂ p )) (cid:3) For the case of arbitrary function A µ ( i∂ p ) the Weyl symbol of F = exp (cid:2) i (ˆ p µ − A µ ( i∂ p )) (cid:3) may be calculatedusing the above obtained expressions. We should represent A in the form of the Laplace transformation: A µ ( x ) = Z (cid:0) ˜ A µ ( k ) e kx + c.c. (cid:1) dk (68)that is A µ ( i∂ p ) = Z (cid:0) ˜ A µ ( k ) e k i∂ p + c.c. (cid:1) dk (69)In turn, this integral may be discretized and represented in the form of the series. As a result the Weylsymbol is given by h exp (cid:16) i ˆ p µ − i Z (cid:2) ˜ A µ ( k ) e k i∂ p + c.c. (cid:3) dk (cid:17)i W = exp (cid:16) ip µ − i A µ ( x ) (cid:17) (70)where A µ ( x ) = Z (cid:2) a µ ( k ) ˜ A µ ( k ) e kx + c.c. (cid:3) dk and a µ ( k ) = 1 − e − k µ k µ e k µ / = sinh( k µ / k µ / .5. Weyl symbol of Wilson Dirac operator ˆ Q ( p − A ( i∂ p ))Now we are in the position to calculate the Weyl symbol of operator ˆ Q for the Wilson fermions in thepresence of arbitrary electromagnetic field A ( x ). For the operatorˆ Q ( p ) = X k =1 , , , γ k g k ( p ) − im ( p ) (72)with g k ( p ) = sin( p k ) m ( p ) = m (0) + X ν =1 (1 − cos( p ν )) (73)we obtain h Q ( p − A ( i∂ p )) i W = X k =1 , , , γ k sin( p k − A k ( x )) − i ( m (0) + X ν =1 (1 − cos( p ν − A ν ( x )))) (74)where A is the following transformation of electromagnetic field: A µ ( x ) = Z (cid:2) sin( k µ / k µ / A µ ( k ) e i kx + c.c. (cid:3) dk (75)while the original electromagnetic field had the form: A µ ( x ) = Z (cid:2) ˜ A µ ( k ) e i kx + c.c. (cid:3) dk
6. Propagator of fermionic quasiparticles in the presence of external electromagnetic field
Let us consider the cubic tight - binding model of Wilson fermions. The two point Green function G ( p , p ) obeys equation ˆ Q ( p − A ( i∂ p ))) G ( p , q ) = δ ( p − q )The Wigner transformation of G is defined as G W ( x n , p ) ≡ Z dqe ix n q G ( p + q/ , p − q/
2) (76)It obeys the Groenewold equation G W ( x n , p ) ⋆ Q W ( x n , p ) = 1 (77) The Gronewold equation has the explicit form1 = G W ( x n , p ) ⋆ Q W ( x n , p ) = G W ( x n , p ) e i (cid:16) ←− ∂ xn −→ ∂ p −←− ∂ p −→ ∂ xn (cid:17) Q W ( x n , p ) (78)Let us introduce the following notation ←→ ∆ = i (cid:16) ←− ∂ x n −→ ∂ p − ←− ∂ p −→ ∂ x n (cid:17) Notice, that the action of this symbol on Q W ( x n , p ) = Q ( p − A ( x n )) is given by Q − ( p − A ( x n )) ←→ ∆ Q ( p − A ( x n )) = − Q − ( p − A ( x n )) i ←− ∂ p j F jk ( x n ) −→ ∂ p k Q ( p − A ( x n ))13here F ij is composed of the transformed gauge potential A of Eq. (75) in the way similar to the fieldstrength: F ij = ∂ i A j − ∂ j A i Let us represent G W ( x n , p ) as the series G W ( x n , p ) = G (0) W ( x n , p ) + G (1) W ( x n , p ) + G (2) W ( x n , p ) + ... where the term G ( i ) W ( x n , p ) has the order i in powers of the derivatives of A with respect to x n . We come tothe following series of equations1 = G (0) W ( x n , p ) Q W ( x n , p )1 = G (0) W ( x n , p ) Q W ( x n , p ) + G (0) W ( x n , p ) ←→ ∆ Q W ( x n , p )+ G (1) W ( x n , p ) Q W ( x n , p )1 = G (0) W ( x n , p ) Q W ( x n , p ) + G (0) W ( x n , p ) ←→ ∆ Q W ( x n , p ) + 12 G (0) W ( x n , p ) ←→ ∆ Q W ( x n , p )+ G (1) W ( x n , p ) Q W ( x n , p ) + G (1) W ( x n , p ) ←→ ∆ Q W ( x n , p )+ G (2) W ( x n , p ) Q W ( x n , p ) ... X i,k =0 ...n ; i + k ≤ n k ! G ( i ) W ( x n , p ) ←→ ∆ k Q W ( x n , p ) (79)From this sequence we obtain G (0) W ( x n , p ) = Q − W ( x n , p ) G (1) W ( x n , p ) = − h Q − W ( x n , p )) ←→ ∆ Q W ( x n , p ) i Q − W ( x n , p ) G (2) W ( x n , p ) = − h G (0) W ( x n , p ) ←→ ∆ Q W ( x n , p ) + G (1) W ( x n , p ) ←→ ∆ Q W ( x n , p ) i Q − W ( x n , p ) ... G ( n ) W ( x n , p ) = − X k =0 ...n − n − k )! h G ( k ) W ( x n , p ) ←→ ∆ k Q W ( x n , p ) i Q − W ( x n , p ) (80)In the other words G (0) W ( x n , p ) = Q − W ( x n , p ) G (1) W ( x n , p ) = − h Q − W ( x n , p )) ←→ ∆ Q W ( x n , p ) i Q − W ( x n , p ) G (2) W ( x n , p ) = − h Q − W ( x n , p ) ←→ ∆ Q W ( x n , p ) − h Q − W ( x n , p )) ←→ ∆ Q W ( x n , p ) i Q − W ( x n , p ) ←→ ∆ Q W ( x n , p ) i Q − W ( x n , p ) ... G ( n ) W ( x n , p ) = − X k =0 ...n − n − k )! h G ( k ) W ( x n , p ) ←→ ∆ n − k Q W ( x n , p ) i Q − W ( x n , p ) (81) We represent the results of this iterative solution as follows: G W ( x n , p ) = X k G ( k ) W ( x n , p ) G ( n ) W ( x n , p ) = X P i =1 ...M k i = n C nk k ...k M h ... h Q − W ←→ ∆ k Q W i Q − W ←→ ∆ k Q W i Q − W ... ←→ ∆ k M Q W i Q − W (82)14et us substitute this expression to Eq. (81): G ( n ) W ( x n , p ) = − X m =0 ...n − n − m )! X P i =1 ...M k i = m C mk k ...k M h ... h Q − W ←→ ∆ k Q W i Q − W ←→ ∆ k Q W i Q − W ... ←→ ∆ k M Q W i Q − W ←→ ∆ n − m Q W i Q − W (83)Constants C nk k ...k M satisfy the following equations, that determine them iteratively: C = 1 C k + ... + k M + k M +1 k k ...k M k M +1 = − k M +1 ! C k + ... + k M k k ...k M (84)We obtain: C = 1 C = − C = − , C , = 1 C = − , C = 12 , C , = 12 , C , , = − C = − , C = 13! , C , = 14 , C , , = − / , C , = 13! , C , , = − , C , , = − , C , , , = 1 ... C k + ... + k M k k ...k M = ( − M k ! k ! ...k M ! ... (85)We obtain the final form of the solution: G W ( x n , p ) = Q − W + X n =1 ... ∞ X M = 1 ...nk + ... + k M = nk i = 0 ( − M k ! k ! ...k M ! h ... h Q − W ←→ ∆ k Q W i Q − W ←→ ∆ k Q W i Q − W ... ←→ ∆ k M Q W i Q − W = X M =0 ... ∞ h ... h Q − W (1 − e ←→ ∆ ) Q W i Q − W (1 − e ←→ ∆ ) Q W i ... (1 − e ←→ ∆ ) Q W i| {z } Q − W M brackets = X M =0 ... ∞ h ... h Q − W (1 − ⋆ ) Q W i Q − W (1 − ⋆ ) Q W i Q − W ... (1 − ⋆ ) Q W i| {z } Q − W M brackets (86)In the first row the sum may be extended to the values M = n = 0, then the first term will be equal to Q − W . Let us introduce the product operator • , which works as follows being combined with the star productintroduced above: A • B ⋆ C = ( AB ) ⋆ C, A ⋆ B • C = ( A ⋆ B ) • C
15n the first equation ⋆ acts both on AB and on C while in the second equation it acts only on A and B .These rules allow to write the above equation in the compact way: G W ( x n , p ) = X M =0 ... ∞ Q − W (1 − ⋆ ) Q W • Q − W (1 − ⋆ ) Q W • Q − W ... (1 − ⋆ ) Q W • | {z } Q − W M • − products = X M =0 ... ∞ (cid:16) Q − W (1 − ⋆ ) Q W • (cid:17) M Q − W (87)We may write symbolically: G W ( x n , p ) = (cid:16) − Q − W (1 − ⋆ ) Q W • (cid:17) − Q − W = (cid:16) Q − W ⋆ Q W • (cid:17) − Q − W (88)The last expression serves also as the alternative proof of Eq. (86) because we started from G W ⋆ Q W = 1.We substitute Eq. (89) to the star product G W ⋆ Q W and obtain G W ⋆ Q W = X M =0 ... ∞ (cid:16) Q − W (1 − ⋆ ) Q W • (cid:17) M Q − W ⋆ Q W = − X M =0 ... ∞ (cid:16) Q − W (1 − ⋆ ) Q W • (cid:17) M Q − W (1 − ⋆ ) Q W + X M =0 ... ∞ (cid:16) Q − W (1 − ⋆ ) Q W • (cid:17) M = − X M =1 ... ∞ (cid:16) Q − W (1 − ⋆ ) Q W • (cid:17) M + X M =0 ... ∞ (cid:16) Q − W (1 − ⋆ ) Q W • (cid:17) M = (cid:16) Q − W (1 − ⋆ ) Q W • (cid:17) = 1 (89)
7. Summary of obtained results and conclusions
In this paper we develop the Wigner Weyl formalism for the model of lattice Wilson fermions. The mainresults specified below are the explicit expression for the Weyl symbol of lattice Wilson Dirac operator andthe explicit expression for the fermion propagator in the presence of arbitrary external electromagnetic field.The main purpose of the work is to provide the necessary tools for the analytical studies of the lattice models.Such a study may precede in many cases the numerical simulations and thus may also improve the latterindirectly. The pure analytical investigation itself of the lattice models is relevant, for example, for the studyof the so - called non - dissipative transport. (It is also called anomalous transport because it reveals thecorrespondence with chiral anomaly, scale anomaly, etc.) In [27, 28, 29, 30, 31, 32] the two first terms (in theexpansion in powers of the derivatives) of the present solution of the Groenewold equation were calculated,which give the linear response of vector/axial currents to the external field strength. It appears that inmany cases the corresponding coefficients are the topological invariants in momentum space, i.e. they arenot changed when the system is changed smoothly. This gives the efficient description of certain anomaloustransport phenomena (chiral magnetic effect, chiral separation effect, anomalous quantum Hall effect, scalemagnetic effect, etc).Our present study generalizes the approach of [27, 28, 29, 30, 31, 32] essentially and gives the morepowerful method of calculations. In the present paper we represent the complete iterative solution of theGroenewold equation (to all orders of the derivative expansion) in the presence of arbitrarily varying externalelectromagnetic field. It will allow to investigate the anomalous transport in case of varying external fieldstrength. We foresee, that the intimate relation between topology and the non - dissipative transport willsurvive here as well, but its description may, possibly, be governed by the more complicated topologicalinvariants.Let us repeat once again the obtained results. The derived expression for the Weyl symbol of the latticeWilson Dirac operator in the presence of arbitrary electromagnetic field appears to be surprisingly simple: h Q ( p − A ( i∂ p )) i W = X k =1 , , , γ k sin( p k − A k ( x )) − i ( m (0) + X ν =1 (1 − cos( p ν − A ν ( x )))) (90)16here by A we denote the following transformation of the original electromagnetic field: A µ ( x ) = Z (cid:2) sin( k µ / k µ / A µ ( k ) e i kx + c.c. (cid:3) dk (91)(The original electromagnetic field itself has the form: A µ ( x ) = R (cid:2) ˜ A µ ( k ) e i kx + c.c. (cid:3) dk .) Eq. (90) is furtherused to calculate the Wilson fermion propagator in the presence of arbitrary electronagnetic field. Namely,we consider first the Wigner transformation G W of the propagator, and solve iteratively the Groenewoldequation. This solution has the form: G W ( x n , p ) = Q − W + X n =1 ... ∞ X M = 1 ...nk + ... + k M = nk i = 0 ( − M k ! k ! ...k M ! h ... h Q − W ←→ ∆ k Q W i Q − W ←→ ∆ k Q W i Q − W ... ←→ ∆ k M Q W i Q − W = X M =0 ... ∞ h ... h Q − W (1 − e ←→ ∆ ) Q W i Q − W (1 − e ←→ ∆ ) Q W i ... (1 − e ←→ ∆ ) Q W i| {z } Q − W M brackets (92)where ←→ ∆ = i (cid:16) ←− ∂ x n −→ ∂ p − ←− ∂ p −→ ∂ x n (cid:17) . The obtained solution for the Wigner transformation of the two pointGreen function allows to reconstruct the Green function itself. In momentum space we have: G ( p + q/ , p − q/
2) = 1(2 π ) D X n e − ix n q G W ( x n , p ) (93)while in coordinate space the propagator is given by˜ G ( z n , y n ) = Z dp (2 π ) D e i ( z n − y n ) p G W (cid:16) z n + y n , p (cid:17) (94)The presented scheme allows to calculate, in principle, the propagator of Wilson fermions on the back-ground of any external electromagnetic field. It would be interesting to apply it, for example, to the cal-culation of the propagator in the presence of varying external magnetic field of the particular form or inthe presence of the particular electromagnetic wave. It is also worth mentioning, that the above obtainedexpressions may be generalized relatively easily to the case, when the external gauge field is non - Abelian.Then, those expressions may be used for the calculation of various observables in QCD using numerical sim-ulations. Many of those observables are expressed through the vacuum average of the products of fermionGreen functions in the presence of dynamical gauge field. We should then first use the expression for thepropagator in the presence of external gauge field. The product of such propagators is represented as a seriesin powers of the derivatives of Q W . In quenched approximation each term should be averaged over the gaugefield with the weight equal to exp( − S ), where S is the pure gauge field action. This is to be done usingnumerical simulations of the pure gauge theory. Depending on the particular problem several simplificationsmay be made: for example, only a few first terms in the series in powers of the derivatives of Q W may betaken. In order to take into account the dynamical fermions we should also include into the consideration thefermion Determinant using the hybrid Monte - Carlo (HMC) algorithm. Our expression for the propagatorin the presence of external field may also become the source of a modification of the HMC algorithm. Butthis is to be the subject of a separate study.Another interesting continuation of the presented research is the extension of the Wigner - Weyl formalismto the more complicated lattice models defined on the rectangular lattices to be used for the regularization ofthe continuum quantum field theory. It would be interesting, in particular, to extend the proposed formalismto the model with overlap fermions that are most suitable for the investigation of QCD in the chiral limit. Weforesee certain difficulties in such an extension related to the calculation of the Weyl symbol of the overlapDirac operator caused by the structure of the latter essentially different from that of the Wilson fermions.The presented results also may be extended to the lattice models with the non - rectangular lattice, thatwill allow to calculate analytically the fermionic quasiparticle propagator in various tight - binding models17f the solid state physics. Unlike the case of the overlap fermions, in this extension in many cases the generalstructure of operator ˆ Q remains similar to that of Eq. (31) with certain new basis matrices γ k instead of theDirac matrices and with the arguments of g and m that depend on the projection of momenta to vectors ofreciprocal lattice.To conclude, in the present paper we propose the extension of the Wigner - Weyl formalism to the latticequantum field theory with Wilson fermions. The proposed formalism gives the efficient algorithm for thecalculation of the Wilson fermions propagator in the presence of arbitrary external electromagnetic field.This technique may be extended in the straightforward way to the wide range of the non - hypercubic latticemodels relevant for the description of the solid state physics. Thus we expect that the proposed techniquemay have applications both to the quantum field theory in lattice regularization and to the solid state physics.Using the lattice Wilson fermion propagator in the presence of arbitrary external electromagnetic field thiswill be possible to investigate via numerical simulations, for example, various effects in quark matter inthe presence of in - homogeneous external electromagnetic field. On the solid state physics side the directapplications will become possible when the similar formalism will be developed for the crystal lattice of morecomplicated and more realistic form. Then our formalism will allow to investigate various phenomena typicalfor the interaction of electronic quasiparticles with photons.Both authors kindly acknowledge useful discussions with Z.V.Khaidukov. Appendix I. Fourier transform
In this Appendix we accumulate for the convenience of the reader the well - known expressions for theFourier transform and Fourier series widely used throughout the text of the paper. For simplicity we considerthe one - dimensional constructions.
Fourier transform: f ( x ) = 1 √ π Z ∞−∞ dke ikx ˜ f ( k ) ˜ f ( k ) = 1 √ π Z ∞−∞ dxe − ikx f ( x ) (95)The integral representation for the delta functions f ( x ) = 12 π Z ∞−∞ dke ikx Z ∞−∞ dx ′ e − ikx ′ f ( x ′ ) = Z ∞−∞ dx ′ f ( x ′ ) 12 π Z ∞−∞ dke ik ( x − x ′ ) (96)hence δ ( x − x ′ ) = 12 π Z ∞−∞ e ik ( x − x ′ ) dk (97)and ˜ f ( k ) = Z ∞−∞ dxe − ikx π Z ∞−∞ dk ′ e ik ′ x ˜ f ( k ′ ) = Z ∞−∞ dk ′ ˜ f ( k ′ ) 12 π Z ∞−∞ dxe − ix ( k − k ′ ) (98)hence δ ( k − k ′ ) = 12 π Z ∞−∞ e − ix ( k − k ′ ) dx (99) x ∈ [0 , L ] - discrete momenta Fourier series: f ( x ) = ∞ X n =0 e ik nx ˜ f ( k n ) k n = k n k = 2 πL ˜ f ( k n ) = 1 L Z L dxe − ik n x f ( x ) (100)18he representation of the delta functions through the integrals/series: f ( x ) = ∞ X n =0 e ik nx L Z L dx ′ e − ik nx ′ f ( x ′ ) = Z L dx ′ f ( x ′ ) 1 L ∞ X n =0 e ik n ( x − x ′ ) (101)hence δ ( x − x ′ ) = 1 L ∞ X n =0 e ik n ( x − x ′ ) (102)and ˜ f ( k n ) = 1 L Z L dxe − ik nx ∞ X m =0 e ik mx ˜ f ( k m ) = ∞ X m =0 ˜ f ( k m ) 1 L Z L dxe − ik x ( n − m ) (103)hence δ ( k n − k m ) = 1 L Z L dxe − ik x ( n − m ) (104) k ∈ [0 , πL ]Fourier series: ˜ f ( k ) = ∞ X n =0 e − ikx n f ( x n ) x n = nLf ( x n ) = 12 π/L Z π/L dke ikx n ˜ f ( k ) (105)The representation of the delta functions as the integrals/series:˜ f ( k ) = ∞ X n =0 e − iknL L π Z π/L dk ′ e ik ′ nL ˜ f ( k ′ ) = Z π/L dk ′ ˜ f ( k ′ ) L π ∞ X n =0 e − inL ( k − k ′ ) (106)hence δ ( k − k ′ ) = L π ∞ X n =0 e − inL ( k − k ′ ) (107)and f ( x n ) = L π Z π/L dke iknL ∞ X m =0 e − ikmL f ( x m ) = ∞ X m =0 f ( x m ) L π Z π/L dke ikL ( n − m ) (108)hence δ ( x n − x m ) = L π Z π/L dke ikL ( n − m ) (109) Appendix II. An expression for the P ordered exponent.
The path-ordering operator is defined as follows P ( f ( x ) g ( y )) ≡ θ ( x − y ) g ( y ) f ( x ) + θ ( y − x ) f ( x ) g ( y ) (110)This definition is consistent with the following equation P e R x x dx f ( x ) ≡ lim ∆ → N Y n =0 e ∆ f ( x + n ∆) (111)where N = x − x ∆ . 19 emma Let ˆ B and ˆ C be operators. Then e ˆ B + ˆ C = P e R due ˆ Bu ˆ Ce − ˆ Bu e ˆ B (112) Proof
After the discretization we come to I = P e R due ˆ Bu ˆ Ce − ˆ Bu e ˆ B = lim ∆ → N = Y n =0 e ∆ e n ∆ ˆ B ˆ Ce − n ∆ ˆ B e ˆ B (113)that is I = h e ∆ e
0∆ ˆ B ˆ Ce −
0∆ ˆ B i h e ∆ e
1∆ ˆ B ˆ Ce −
1∆ ˆ B i ... h e ∆ e (1 − ∆) ˆ B ˆ Ce − (1 − ∆) ˆ B i h e ∆ e ˆ B ˆ Ce − ˆ B i e ˆ B (114)Next, using relation e ˆ O ˆ A ˆ O − = 1 + ˆ O ˆ A ˆ O − + ˆ O ˆ A ˆ O − ˆ O ˆ A ˆ O − + ... = ˆ Oe ˆ A ˆ O − , we may rewrite I as follows I = h e ∆ ˆ C i h e ∆ ˆ B e ∆ ˆ C e − ∆ ˆ B i h e ∆ ˆ2 B e ∆ ˆ C e − ∆ ˆ2 B i ... h e (1 − ∆) ˆ B e ∆ ˆ C e − (1 − ∆) ˆ B i h e ˆ B e ∆ ˆ C e − ˆ B i e ˆ B = e ∆ ˆ C h e ∆ ˆ B e ∆ ˆ C i N ≈ (1 + ∆ ˆ C ) h (1 + ∆( ˆ B + ˆ C ) + ∆ ˆ B ˆ C ) i N ≈ h (1 + ∆( ˆ B + ˆ C ) i N = e ˆ B + ˆ C (115)Let us also prove the following Lemma
P exp (cid:20) i Z u u e ipu A ( i∂ p ) e − ipu du (cid:21) e − ipx = exp i u − u Z A ( x + u ) du − ipx (116) Proof
20 exp (cid:20) i Z u u e ipu A ( i∂ p ) e − ipu (cid:21) e − ipx = lim ∆ → N = u − u Y n =0 e i ∆ e ipn ∆ A ( i∂ p ) e − ipn ∆ e − ipx = lim ∆ → N = u − u Y n =0 e ipn ∆ e i ∆ A ( i∂ p ) e − ipn ∆ e − ipx = h lim ∆ → (cid:16) e i ∆ A ( i∂ p ) (cid:17) (cid:16) e ip ∆ e i ∆ A ( i∂ p ) e − ip ∆ (cid:17) ... (cid:16) e ipN ∆ e i ∆ A ( i∂ p ) e − ipN ∆ (cid:17)i e − ipx = h lim ∆ → (cid:16) e i ∆ A ( i∂ p ) (cid:17) (cid:16) e ip ∆ e i ∆ A ( i∂ p ) (cid:17) ... (cid:16) e ip ∆ e i ∆ A ( i∂ p ) (cid:17) e − ip ( u − u ) i e − ipx =lim ∆ → (cid:16) e i ∆ A ( i∂ p ) (cid:17) (cid:16) e ip ∆ e i ∆ A ( i∂ p ) (cid:17) N e − ip ( x +( u − u )) =lim ∆ → (cid:16) e i ∆ A ( i∂ p ) (cid:17) (cid:16) e ip ∆ e i ∆ A ( i∂ p ) (cid:17) N − e ip ∆ e i ∆ A ( x +( u − u )) e − ip ( x +( u − u )) =lim ∆ → (cid:16) e i ∆ A ( i∂ p ) (cid:17) (cid:16) e ip ∆ e i ∆ A ( i∂ p ) (cid:17) N − e − ip ( x +( u − u ) − ∆) e i ∆ A ( x +( u − u )) =lim ∆ → (cid:16) e i ∆ A ( i∂ p ) (cid:17) (cid:16) e ip ∆ e i ∆ A ( i∂ p ) (cid:17) N − e − ip ( x +( u − u ) − e i ∆( A ( x +( u − u ) − ∆))+ A ( x +( u − u )) =lim ∆ → exp i N = u − u X n =0 ∆ A ( x + n ∆) e − ipx = exp i u − u Z duA ( x + u ) − ipx (117) Appendix III. BCH formula for the particular Lie algebra
In this section we derive the particular case of the BCH formula, that corresponds to the Lie algebra ofoperators composed of the basis elements X , Y with the commutation relation [ X, Y ] = αY . We are goingto prove the following Lemma e X e Y = exp( X + α − e − α Y ) (118)or, alternatively exp( X + Y ) = e X e βY (119)where β = − e − α α . Proof