Classical relativistic systems of charged particles in the front form of dynamics and the Liouville equation

Classical relativistic system of point particles coupled with an electromagnetic ﬁeld is considered in the three-dimensional representation. The gauge freedom connected with the chronometrical invariance of the four-dimensional description is reduced by use of the geometrical concept of the forms of relativistic dynamics. The remainder gauge degrees of freedom of the electro-magnetic potential are analysed within the framework of Dirac’s constrained Hamiltonian mechanics in the front form of dynamics. The results are implemented to the problems of relativistic statistical mechanics. Based on the corresponding Liouville equation the classical partition function of the sys-tem is written down in a gauge-invariant manner and an integration over ﬁeld variables is performed.


Introduction
All fundamental interactions in physics have got gauge nature and demand the use of singular Lagrangians.An adequate method for dealing with such systems was developed by Dirac (see [1,2,3]).His approach, i.e., constrained Hamiltonian mechanics, has been elaborated in many directions [4,5] and usefully applied to the problems of quantum field theory (see, e.g., [6,7]).But an application of this technique to the relativistic statistical mechanics [8,9] seems rather obscure, although, for example, the thermo field formulation of the condensed matter theory [10] admits a variational approach with the same gauge structure as in quantum electrodynamics and in related theories.We only note the paper [11], where the use of Faddeev' measure [6] was suggested for the construction of the relativistic partition function of directly interacting particle system described by means of the constrained Hamiltonian formalism.
The present paper is concerned with the classical relativistic system of point charges coupled with an electromagnetic field.The first attempt to analyse the constraint contents of such a system was made by Dirac [12].The corresponding action functional in the four-dimensional representation has got two kinds of the gauge freedom.The first is connected with arbitrariness in the parametrisation of particle world lines (chronometrical invariance); the second is generated by the proper gauge transformations of electromagnetic potentials.We reduce the gauge freedom of the first kind by means of the farther great Dirac's invention, namely, the concept of the forms of relativistic dynamics [13].Here we use the notion of the form of dynamics as denoting the description of relativistic system, which corresponds to a given global simultaneity relation defined by means of some foliation of the Minkowski space by space-time or isotropic hypersurfaces.Moreover, we consider only the case when the corresponding simultaneity relation is independent of particle or field configurations (cf.[14]).Using Dirac's constraint formalism, we try to isolate the gauge degrees of freedom and to formulate the statistical description of the system in a gauge-invariant manner.
The general features of the formalism were presented in [15] within the instant form of dynamics.It will be our purpose in the present paper to explore the further possibilities connected with other forms of relativistic dynamics; in particular, we shall consider the front form description of the charged particle system.
The paper is organised as follows.In Sec. 2 we will first establish the structure of Hamiltonian description of the system of particles plus field in an arbitrary form of relativistic dynamics.Our results are then applied to the front form of dynamics.In Sec. 3 we perform the analysis of the corresponding constraints.Section 4 contains the elimination of the gauge degrees of freedom by suitable canonical transformation.Sections 5 and 6 are devoted to the application in relativistic statistical mechanics.We formulate a Liouville equation for distribution function in the front form of dynamics and write down its equilibrium solution corresponding to classical Gibbs ensemble accounting particle and field degrees of freedom.Peculiarity of the front form of dynamics allow us to perform an integration over field variables in a relativistic partition function.Some calculations of a purely technical nature are collected in appendices.

Charged particle system in an arbitrary form of dynamics
We shall consider a system of N charged particles, which is described by their (timelike) world lines in the Minkowski space-time1 addtoresetequationsection An interaction between charges is assumed to be mediated by an electromagnetic field F = dA with the electromagnetic potential 1-form The dynamical properties of such a system are completely determined by the following action functional [16,17,18] ) where m a and e a denote the mass and the charge of particle a, respectively, u µ a (τ a ) = dx µ a (τ a )/dτ a .We are interested in constructing the Hamiltonian description of the system in a given form of relativistic dynamics.The particle and field degrees of freedom will be treated on equal level.
The form of relativistic dynamics in its geometrical definition is specified by the foliation Σ = {Σ t | t ∈ R} of the Minkowski space-time with space-like or isotropic hypersurfaces (see [19,20,21]).As it follows from condition (2.6), ∂ 0 σ = 0 and therefore the hypersurface equation (2.5) can be solved with respect to x 0 in the form: For definiteness we put Making use of the identities it is easy to see that condition (2.6) implies the following inequality for the function ϕ: (2.10) Then one can consider the particle world lines γ a as being determined by a set of points x a (t) = γ a Σ t of intersections with the elements of the foliation Σ.The parametrical equation of the world line in a given form of dynamics is (2.11) The variable t serves as a common evolution parameter of the particle system.Therefore, the choice of the form of dynamics is equivalent to the fixing of the parameters τ a = t of the particle world lines in the reparametrization-invariant action (2.4).
On the other hand, the foliation Σ specifies a certain 3 + 1 splitting of the Minkowski space-time M 4 ∼ = R × Σ 0 , because by the definition of a foliation all hypersurfaces Σ t for different t are diffeomorphic to Σ 0 .That splitting, f : (2.12) Accounting that the determinant of the transformation (2.12) is one can immediately rewrite the action functional (2.4) into the form with the Lagrangian function The dynamical variables of our variational problem will be the functions x a (t), A µ (t, x) and their first order derivatives with respect to the evolution parameter, v a (t) and A µ ,t (t, x).Using the obvious relations we find the following expression for the field Lagrangian density: where The canonical momenta of our problem are given by The basic Poisson brackets are all other brackets vanish.Equations (2.21) and (2.22) imply the primary constraint (2.24) In the isotropic forms of dynamics, however, some additional primary constraints may occur.To see that, we rewrite equation (2.21) into the form where S i does not contain the derivatives with respect to the evolution parameter t, and (2.27) Upon using the identity β ij ϕ j = ϕ i , one obtains equation (2.28) The matrix In the case of space-like forms of dynamics (ϕ 2 < 1) the matrix is nonsingular and possesses an inverse so that equation (2.28) can be solved as For isotropic forms of dynamics we have β ij = ϕ i ϕ j and (2.28) implies constraints Because it holds identically ϕ i ξ i = 0, really only two additional primary constraints occur.
Next, we consider the canonical Hamiltonian of our system, which is defined as (2.32) The immediate calculations give and The problem of solution of (2.20) with respect to the particle velocities is analogous to the case of free particles [22].For completeness we collect the corresponding expressions in Appendix A. Replacing into (A.9)p i by (2.36) yields the one-particle Hamiltonian for the space-like forms of dynamics: Similarly, for the isotropic forms of dynamics we obtain from (A.11) To express the field Hamiltonian (2.35) in the terms of canonical variables we rewrite it into the form (2.39) Then for the space-like forms of dynamics, taking into account (2.29) and (2.30), we obtain In the case of isotropic forms of dynamics (ϕ 2 = 1), when β ij = ϕ i ϕ j , (2.28) gives so that (2.39) takes the form Detailed analysis of the Hamiltonian description for the charged particle system in the instant form of dynamics was carried out in [15] with the application to the classical relativistic statistical mechanics.In the following sections we shall be concerned with the front form of dynamics, analyzing additional primary constraints and ensuring the corresponding Liouville equation.

Constraint analysis in the front form of dynamics
Let us consider the family of the forms of dynamics, which is given by According to (2.24), (2.31), the set of primary constraints is It is easy to see that (3.3) is equivalent to where we define orthogonal and longitudinal projections of an arbitrary 3-vector f k with respect to the vector n k as The canonical Hamiltonian of the system is determined by Then we get Dirac Hamiltonian which takes into account primary constraints (3.2) and (3.4): where λ, λ k are the Dirac multipliers.
The preservation of the constraint (3.2) in time produces the secondary constraint: where ≈ means "weak equality" in the sense of Dirac and is a charge density.Next, we consider commutation relations between constraints (3.2), (3.4), and (3.8): We can check directly, that Therefore, the two constraints belong to the first class.Taking into account (3.12), we come to the conclusion that the constraints (3.4) are of the second class.However, there are two independent second class constraints only.Then we can reduce them by means of Dirac bracket: where C αβ (x − y) is an inverse matrix to Ω αβ (x − y): In the next chapter we shall consider the elimination of the first class constraints and the formulation of the Hamiltonian description in the terms of independent physical variables.Now let us canonically transform the field variables: After the transformation the set of constraints of our system becomes where now k ai = p ai − e a A i (t, x a ).

Elimination of the gauge degrees of freedom
Let us consider the first class constraints.We see immediately that A 0 , E are a pair of conjugated gauge canonical variables.A second such pair is formed by . Therefore, we can separate the gauge degrees of freedom and gauge-invariant ones by means of the Hodge decomposition (see, e.g., [23,14]): where Then we have Now we have to do a canonical transformation to the new variables ((x i a , πai ), (a α , ẽα ), (Q, Γ), (A 0 , E)), which is generated by the functional We obtain The transverse part of A i , is connected with the new canonical variables as Now it is convenient to take n = (0, 0, 1) and perform the following canonical transformation of the field and particle momenta: The set of constraints for our system in the terms of the new variables can be written as follows: (E, Γ, e α − a α,3 ) ≈ 0. (4.10) We reduce the second class constraints by means of the Dirac bracket: Here C αβ (x − y) is an inverse matrix to ) It is given by because C αβ (x − y) must be antisymmetric.Let us introduce the following denotation: The final form of the Dirac bracket is where {F, G} (x,π) denote the standard Poisson bracket in the terms of x and π.Elimination of constraints (4.10) into H c leads to the physical Hamiltonian: which generates evolution of an arbitrary function f depending on the gauge-invariant variables x i a , π ai and a α , e α in the terms of the Dirac bracket (4.16): The gauge-invariant volume element of the constrained field phase space (a α , e α ) is written as where γ is defined as a normalisation constant of the Gauss integral [7]: Taking into account (4.19), we can write the volume element of the physical phase space of the described system: Here is the volume element of the particle phase space.Now we need check the Liouville theorem: Γ ph (t) = Γ ph (t 0 ).It is well known from the classical mechanics, that system evolution in the phase space can be described by means of a canonical transformation in the terms of the Poisson bracket, which immediately leads to the conservation of the phase space volume.This proves the conservation of Γ p ph .In our case the field evolution is generated by the Dirac bracket.Nevertheless, the conservation of the volume of the constrained field phase space can be proved and we shall demonstrate its possible proof for the considered Dirac bracket (4.16)However, we turn off dΓ g (t) from the further description, because all thermodynamical characteristics of the system do not depend on dynamics of the gauge degrees of freedom (see [15]).

Statistical mechanics
Now let us imagine that the physical initial data x i a (t 0 ), π ai (t 0 ), a α (t 0 , x), e α (t 0 , x) are not precisely known.Hence, we can introduce a probability density ̺(t 0 ) ≡ ̺(t 0 , x i a (t 0 ), π ai (t 0 ), a α (t 0 , x), e α (t 0 , x)) for having various initial states.This function satisfies the condition ̺(t 0 )dΓ ph (t 0 ) = 1. (5.1) Then average value of a general dynamical variable f is defined by We shall find below the value of Z.
Let us first rewrite the physical Hamiltonian as follows where Since H ph does not depend on field momenta e α , after an integration over e α in (5.6) we obtain the expression for the partition function: Here we can replace da α by dã α .Taking into account (4.8), the integration over ãα yields where Det − βδ αβ ∆δ 3 (x − y) (5.13) represents the free field partition function.

Conclusions
In this paper we have once more demonstrated the usefulness of the various forms of relativistic dynamics in treating the relativistic particle systems.Constraint analysis of the considered problem shows significantly different structures of Hamiltonian description of charged particles, interacting by means of an electromagnetic field, in the space-like and isotropic forms of dynamics.Specifically, the use of front form of relativistic dynamics allows us to exclude the electromagnetic field variables from classical partition function of the system of charged particles.It is obvious that such an exclusion must depend on the particular boundary conditions for the field variables, but at present the proper sense of these conditions remains unclear.The obtained representation for the partition function contains highly nonlocal expressions, and their further analysis consist of a complicated task.The consideration of the various approximation schemes at this step seems to be inevitable.It should be noted that exclusion of the field variables transform the problem into the domain of relativistic direct interaction theory (see, e.g., [24,25]).An application of such a theory to the consistent formulation of the relativistic statistical mechanics is just at the beginning.
On the other hand, the established Liouville equation may be used in nonequilibrium situation as well.It serves an useful starting point for deriving various new forms of kinetic equations for charged particle system.namely, the time evolution preserves the phase space volume Γ ph .