Hamiltonian Dynamics Of Condensed Media With Spontaneously Broken Phase Invariance

The Hamiltonian formalism to the dynamics of the condensed media with spontaneously broken phase invariance has been built. The method for nding of Poisson brackets for the dynamic variables has been formulated. It is based on the consideration of the transformations that leave invariant the kinematic part of the action and on the interpretation of the outside integral terms as the generators of these transformations. Important role in the developed approach plays the construction of the translation operators and their densities for the diierent physical elds, guring in the action. The order parameter phases that correspond to the broken symmetry are deened in terms of the quantities canonically conjugate to the additive motion integral densities. In capacity of speciic objects the quantum crystal, the su-perruid phase of 3 He{B and the quantum spin crystal are considered. The determination of the Poisson brackets (PB) for dynamic variables plays a principal role in the Hamiltonian approach to the theory of various physical systems. In 1] PB were derived on the basis of consideration of the variables' variations at transformations of the group corresponding to the broken symmetry of the system. In the suggesting approach it is shown how the PB for diierent physical systems with broken phase invariance can be obtained proceeding from the transformations that leave invariant the kinematic part of the action (or Lagrangian). The outside integral terms in the action variation are interpreted as the generators of these transformations. Because the setting of the kinematic part of the Lagrangian plays a principal role in our approach we stop in detail on the way of construction of the kinematic part. An important part of this procedure is the construction of the translation operators and their densities for diierent physical elds guring in the action. As an examples of the speciic physical objects the dynamics of the quantum crystal, superruid liquids and the quantum spin crystal are studied.

The determination of the Poisson brackets (PB) for dynamic variables plays a principal role in the Hamiltonian approach to the theory of various physical systems.In 1] PB were derived on the basis of consideration of the variables' variations at transformations of the group corresponding to the broken symmetry of the system.In the suggesting approach it is shown how the PB for di erent physical systems with broken phase invariance can be obtained proceeding from the transformations that leave invariant the kinematic part of the action (or Lagrangian).The outside integral terms in the action variation are interpreted as the generators of these transformations.Because the setting of the kinematic part of the Lagrangian plays a principal role in our approach we stop in detail on the way of construction of the kinematic part.An important part of this procedure is the construction of the translation operators and their densities for di erent physical elds guring in the action.As an examples of the speci c physical objects the dynamics of the quantum crystal, super uid liquids and the quantum spin crystal are studied.
In the frame of the Hamiltonian approach it is easy to formulate the di erential conservation laws connected with the di erent symmetry properties of Hamiltonian.Equation of motion for density a(x) of arbitrary physical quantity A = R d 3 x a(x) can be represented, accordingly to 2], in the form _ a(x) = fa(x); Hg fA; "(x)g ?@a k (x) @x k ; (1:8) where a k (x) = Z d 3 x 0 x 0 k Z 1 0 d fa(x + x 0 ); "(x ?(1 ?)x 0 )g: If quantity A is the generator G of the symmetry group of Hamiltonian then equation (1.8) has the form of the di erential conservation law.

PB for variables of the classical continuum
At rst we consider the way of the construction of the kinematic part of Lagrangian for the classical continuum and the derivation on its basis of the PB of the continuum variables.It is useful for illustration how the di erent translation operators of physical elds are introduced in the theory.We start from the kinematic part 2] L k (x) = i (x)b ?1 ij (x) _ u j (x); b ij ij ?r j u i ; (2:1) where i (x) is the density of momentum, u i (x) is the displacement vector.With the use of the kinematic part (2.1) PB for variables u i (x); i (x) may be obtained.For the derivation of the adiabaticity equation it is necessary to know also the PB of the entropy density (x) with other variables.For nding these PB we write the Lagrangian kinematic part in the form L k (x) = p j (x) _ u j (x) ?(x) _ (x); (2:2) where p j (x) = ( i (x) ?(x)r i (x))b ?1 ij (x): Variable (x) which is conjugated to variable (x) has been introduced in the kinematic part formally and at the obtaining of the motion equation will be considered as cyclic.Let us explain the structure of the quantity p j .Note that the density of momentum i (x) in (2.1) is connected with the translations in the space of variables u i (x); i (x).After the new variables (x); (x) have been introduced the density of momentum i (x) is connected already with the translations in the total space of variables u i (x); i (x); (x); (x) and can be represented in the form i (x) = ~ i (x) + i (x); where ~ i (x) is the momentum density connected with the translations only in the space of variables u i (x) and i (x), and i (x) is connected with translations in the space of (x); (x).Because the quantities (x) and ?(x) are the generalized coordinates and momenta then it follows i (x) = (x)r i (x).Hence, the momentum density ~ i (x) connected with the translations in the space of variables u i (x); i (x) has the form ~ i (x) = i (x) ?(x)r i (x) and we obtain the structure of the quantity p j in (2.2).It is easy to see that variations where functions f i (x); (x) do not depend on u i (x); p i (x); (x); (x), leave invariant the kinematic part L k = R dx L k (x) and may be represented as (2:4) Here G is the generator of the transformations (2.3) which according to (2:5) From (2.4), (2.5) it follows that fu i (x); p j (x 0 )g = ij (x ?x 0 ); f (x); (x 0 )g = (x ?x 0 ): (2:6) Let us nd now the PB for variables u i (x); p i (x); (x) and (x).It is easy to convince that from formulae f~ i (x); (x 0 )g = f~ i (x); (x 0 )g = 0; where ~ i (x) i (x) ?(x)r i (x) = p j (x)b ji (x); follow the brackets f i (x); (x 0 )g = ?(x)r i (x ?x 0 ); f i (x); (x 0 )g = r i (x) (x ?x 0 ): With the taking into account of (2.6) for the quantities i (x) (x)r i (x) we have f i (x); k (x 0 )g = k (x)r 0 i (x ?x 0 ) ? i (x 0 )r k (x ?x 0 ): From this formula and from the bracket fp i (x); p k (x 0 )g = 0 we obtain f i (x); k (x 0 )g = k (x)r 0 i (x ?x 0 ) ? i (x 0 )r k (x ?x 0 ): (2:7) The rst formula in (2.6) leads to the equality fu i (x); k (x 0 )g = b ik (x) (x ?x 0 ).Thus nontrivial PB for variables of continuum have the form f (x); (x 0 )g = (x ?x 0 ); f i (x); (x 0 )g = ?(x)r i (x ?x 0 ); f i (x); (x 0 )g = r i (x) (x ?x 0 ); fu i (x); k (x 0 )g = b ik (x) (x ?x 0 ) ; f i (x); k (x 0 )g = k (x)r 0 i (x ?x 0 ) ? i (x 0 )r k (x ?x 0 ): (2:8) Dynamic equations of continuum 2] can be obtained with the use of (2.8) if we take into account the general structure of the system Hamiltonian H = Z d 3 x "(x); "(x) = " x; (x 0 ); i (x 0 ); b ij (x 0 ) Here "(x) is the energy density which is, in general case, some functional of variables (x); i (x); b ij (x) (variable is cyclic).Note that due to the property of invariance of "(x) with respect to the translations of the Euler variables the density of energy "(x) does not depend on the quantities u i (x), but only on their derivatives @u i @x k or, it is the same, on the quantities b ik (x).

Quantum crystal
Now we shall study the super uid systems for which the ground state is characterized by the spontaneously broken symmetry with respect to the phase transformations.At rst we consider the quantum crystal.Phenomena of it super uidity 3] is connected with the possibility of two kinds of motion.The rst one is the motion of lattice points as in solids, the second one is the transfer of mass by the quasiparticles under the condition of the xed lattice points as in the super uid liquid.
Variables which describe the violation of phase and translational invariances of the quantum crystal ground state are the super uid phase (x) and the displacement vector u i (x) in the con gurational space.Therefore the energy density "(x) is, in general case, the functional of the entropy (x), mass %(x), and momentum i (x) densities and of the super uid phase (x) and displacement vector u i (x): "(x) = " x; (x 0 ); %(x 0 ); i (x 0 ); (x 0 ); u i (x 0 ) : Now, let us obtain the PB of the dynamic variables.We write the density of the Lagrangian kinematic part in the form where ~ i = i ?r i ?%r i : Expression (3.1) is built analogously to the case of the classical continuum (see the explanation of the introduction of term _ ).Namely, the last term in (3.1) represents itself the account of super uidity; correspondingly, in the density momentum ~ i , which is connected with translations in the space of variables u i ; i , it is necessary to subtract additional term %r i .For derivation of PB, as in the case of classical continuum, the variations (2.3) with momentum p j = ~ i b ?1 ij as well as variations %(x) = 0; (x) = g(x) with generator G = Z d 3 x fp i (x)f i (x) ?(x) (x) ?%(x)g(x)g should be cousidered.Making further computations analogously to Section 2 we come to the following algebra of the PB for variables of quantum crystal f i (x); u k (x 0 )g = ?(ik ?r i u k (x)) (x ?x 0 ); f i (x); (x 0 )g = ?(x)r i (x ?x 0 ); f i (x); %(x 0 )g = ?%(x)ri (x ?x 0 ); f i (x); (x 0 )g = (x ?x 0 )r i (x); f (x); (x 0 )g = f%(x); (x 0 )g = (x ?x 0 ); f i (x); k (x 0 )g = k (x)r 0 i (x ?x 0 ) ? i (x 0 )r k (x ?x 0 ); f i (x); (x 0 )g = r i (x) (x ?x 0 ): (3:2) Further variable will be considered as cyclic.Note that due to the invariance of "(x) with respect to the phase transformations and space translations the density "(x) depends not on quantities (x); u i (x) themselves, but only on their derivatives r i p i ; r k u i (or, in the last case, on the quantities b ik = ik ?r k u i ): "(x) = " x; (x 0 ); %(x 0 ); i (x 0 ); p i (x 0 ); b ik (x 0 ) : Vector p has the sense of the super uid momentum.Considering that energy density is the function of the dynamic variables at point x we write now the dynamic equations in the local form.We use for this the results of the rst section.Putting in formula (1.8) consistently a(x) = %(x); "(x); i (x) , and supposing the invariance of the energy density "(x) with respect to the phase and translation transformations, we obtain j k = % @" @ k + @" @p k ; t ik = p ik + i @" @ k + p i @" @p k + b ji @" @b jk ; q k = @" @ k @" @ + % @" @% + l @" @ l + @" @p k @" @% + p i @" @ i +b ij @" @b ik @" @ j ; (3:3) p ?" + @" @ + l @" @ l + % @" @% : Here p is the pressure and j k ; q k ; t ik are the uxes densities of the mass, energy and momentum, respectively.These equations coincide with the obtained earlier 5] on the basis of microscopic approach.Note that formulae (3.3) for the densities uxes can be written in the compact form by introducing the density of the thermodynamic potential != Y a a ?; (3:4) where Y a are the thermodynamic forces de ned by the equalities @" @ 1 Y 0 ; @" @ i ?Y i Y 0 ; @" @% ?Y 4 Y 0 : (3:5) From formulae (3.4),(3.5)follows that @! @Y a = a and, since != !(Y a ; p i ; b ik ), the second principle of thermodynamics in terms of ! has the form d! = a dY a + @! @p i dp i + @! @b ik db ik : With the account of equations (3.3), (3.6) we present the uxes densities in the form ak = ?@ @Y a !Y k Y 0 + @! @p k @ @Y a Y 4 + Y i p i Y 0 + @! @b jk @ @Y a b jl Y l Y 0 : (3:7) Here 0k = q k ; ik = t ik ; 4k = j k .Dynamics of the additive motion integral densities and of the parameters, describing the broken symmetry, is de ned according to (3.7) by equations

Super uid 4 He
On the ground state of the super uid 4 He the phase invariance is broken.
The energy density "(x) is the functional of the entropy (x), mass %(x) and momentum i (x) densities as well as of the super uid phase (x) playing the role of the order parameter: "(x) = " x; (x 0 ); %(x 0 ); i (x 0 ); (x 0 ) .Algebra of the super uid 4 He dynamic variables forms the subalgebra of the quantum crystal algebra (3.2).Hence, uxes densities and equations of motion have the form (3.3), (3.8)where it is necessary to put down the terms containing quantity b ik .This fact corresponds to the one that variable u i is cyclic.

Quantum spin crystal
At consideration of the quantum crystal dynamics we did not took into account the in uence of the spin degrees of freedom that can be essential for quantum solid 3 He.Taking into account, that in the super uid liquid phases of 3 He the symmetry with respect to the uniform spin rotations is broken, we consider the case of the total violation of the spin invariance.
The parameter describing such breaking is the real rotation matrix that corresponds to the B{phase of 3 He.Dynamical variables of the quantum spin crystal with the broken symmetry with respect to the spin rotations are the entropy (x), mass %(x) and spin s (x) densities as well as the orthogonal matrix of rotation a (x), super uid phase (x) and the displacement vector u i (x).Therefore the energy density "(x) is, in general case, presented as a functional "(x) = " x; (x 0 ); %(x 0 ); i (x 0 ); s (x 0 ); a (x 0 ); (x 0 ); u i (x 0 ) : The density of the Lagrangian kinematic part for the quantum spin crystal is written in the form L k (x) = ~ i (x)b ?1 ij (x) _ u j (x) ?(x) _ (x) ?%(x) _ (x) ?s (x)! (x); (5:1) where ~ i = i ?r i ?%r i ?s !k ; != 1 2 a _ a ; !k = 1 2 a r k a : Here the last term in (5.1) is the spin kinematic part of Lagarangian 4,6], the term s !k in quantity ~ i is the corresponding translation operator in the space of variables s ; a such that ~ i is as ordinary the translation operator in the space of u i ; i .If we consider, in the same way as above, the variations of the dynamic variables leaving invariant L k (x) then the algebra for the PB of dynamic variables is obtained which contains the algebra (3.2) and additional brackets 4] f i (x); s (x 0 )g = ?s(x)r i (x ?x 0 ) ; f i (x); a (x 0 )g = (x ?x 0 )r i a (x); fs (x); s (x 0 )g = s (x) (x ?x 0 ); fs (x); a (x 0 )g = a (x) (x ?x 0 ): (5:2) Considering that the energy density is of the following form "(x)=" (x); %(x); i (x); p i (x); b ik (x); s (x); a (x); !k (x) we obtain dynamic equations in the local limit.Using formula (1.8) in which a(x) = f%(x); s (x); "(x); i (x)g and supposing that "(x) obeys the properties of the phase, translational and spin invariances we nd for the uxes densities of mass, spin, energy and momentum j k = % @" @ k + @" @p k ; j k = s @" @ k + @" @! k ; q k = @" @ k @" @ + % @" @% + l @" @ l + s @" @s + @" @p k @" @% + p i @" @ i + + @" @! k @" @s + !i @" @ i + b ij @" @b ik @" @ j ; (5:3) t ik = p ik + i @" @ k + p i @" @p k + !i @" @! k + b ji @" @b jk : For the compact entry of the uxes densities (5.3) we introduce thermodynamic potential != Y a a ?; a = (0; i; ; 4); a = ("; i ; s ; %); (5:4) where Y a are the thermodynamic forces, de ned according to the equalities @" @ 1 Y 0 ; @" @ i ?Y i Y 0 ; @" @s ?Y Y 0 ; @" @% ?Y 4 Y 0 : (5:5) With the account of (5.4), (5.5) we have @!@Y a = a and the second principle of thermodynamics for the Gibbs potential is written as d! = a dY a + @! @p i + @! @b ik db ik + @! @! k d! k : (5:6) It follows from equations (5.3), ( Here 0k = q k ; ik = t ik ; k = j k ; 4k = j k .Dynamics of the additive motion integral densities and of the parameters describing violated symmetry is de ned by equations _ a = ?rk ak ; _ a = 1 Y 0 (Y i r i a ?Y a ) ; (5:7)

Super uid 3 He{B
In the case of the super uid 3 He{B phase the symmetry with respect to the phase transformations and spin rotations is spontaneously broken.The parameters describing such violation are the super uid phase (x) and the orthogonal rotation matrix a (x).Accodingly, the energy density is the functional of the form "(x) = " x; (x 0 ); %(x 0 ); i (x 0 ); s (x 0 ); a (x 0 ); (x 0 ) .
PB for the dynamic variables of the super uid 3 He{B phase form the subalgebra of the PB algebra (3.2),(5.2) for the quantum spin crystal.Thus, uxes densities and equations of motion have the form (5.3),(5.7)where it is necessary to put down terms containing quantity b ik (variable u i is cyclic).

Conclusion
The Proposed approach for the construction of the dynamic variables PB permits us to study from the unique point of view di erent physical systems with spontaneously broken phase invariance.A principal role in the given method plays the structure of the kinematic part of Lagrangian L k and the variations of dynamic variables that leave invariant the kinematic part L k .
The density of entropy is considered as a dynamic variable that leads to the necessity to introduce the corresponding conjugated variable which should be considered as cyclic.Further development of the present formalism can be connected with the introducing into the theory of the di erent gauge elds that corresponds to the consideration of the various types of defects in the systems studied above.