- Email: [email protected]
On the statistical mechanics of a relativistic quantum plasma.
Physica 31 1599-1602
Balescu, R. 1965
LETTER TO THE EDITOR On the statistical mechanics
of a relativistic
quantum plasma.
Although the problem deserves a great interest, it does not seem that the quantum statistical mechanics of relativistic systems has been developed in a systematic way, especially out of equilibrium. In particular the study of a relativistic electron-positron gas interacting with radiation is particularly interesting as it should lead to such important issues as the rigorous derivation of the laws of macroscopic electrodynamics, the hydrodynamics of a gas of Dirac particles, etc. We have started a program on this vast field, and we wish to sketch in the present letter the first results, giving the framework of the theory. We consider a system corresponding to a classical set of N electrons. As, however, the number of particles is not conserved in a relativistic system, the specification of the system has to be made by its charge, which is a well-defined concept. We therefore assume the system to be composed of N charge entities. This system can be realized by N, electrons and N, positrons, with N, - N, = N = const. The system is assumed to be imbedded in a neutralizing background of positive charge. The electromagnetic field produced by the charges is described by a set of transverse oscillators and by the electrostatic Coulomb interactions (Coulomb gauge). Let I@; t) be a solution of the Dirac one-particle equation; we expand it into a set of plane waves normalized in a cubic box of volume Q, and apply the second quantization, obtaining $a(%) = Q-a x { x [~a(&, S) b,,s + v*a(k; s) d~k,s]}.eik’r k s=fl (1) where ua(k, s)[va(k, s)] are plane-wave solutions of the Dirac equation, corresponding to wave vector k, spin s and positive [negative] energy; bks[dks] is the destruction operator for an electron [positron]. 4a(Rk) is defined by the quantity in curly brackets in the first line of eq. (1) : it is a 4-spinor consisting of a superposition of electron and positron states, playing the role of a “quasi particle”, which might be called a “chargeon”. This interpretation is supported by the fact that the well-known anticommutation relations for b and d imply the following relations: [P(k),
++@(k')l+ = 6aS&k,;
[W4, PWl+ The electromagnetic
= L++"(k), ++W')l+ = 0.
(4
vector potential is expanded in the usual way : (3)
where eL is a polarization unit vector, and I denotes the set of a wave vector KA and a polarization index ~11= 1, 2 (KA -6~ = 0 in Coulomb Gauge). The creation and
-
1599 -
1600
R. BALESCU
destruction
operators
a;,
an obey
the usual commutation
relations.
The hamiltonian
is then:
= x (RccP-k
H,
+ ,P’mc~) g+p(iik) ,$a(Rk)
k
H int =
Q-t I: I; (IIRc/K~)~ ~PY~AC#J +P(hk)
-e
kA P(ltk
The coulomb vistic
part is not written
statistical
If A(x) is regarded variables,
as a classical
as follows
now want
(a;
a description
characterized
representation
quantity
because
it is treated
system
fW(PQ)
(i.e. a c-number),
as in non-relati-
and angle
variables
+ aA) = QA =
(nc~,$+q$ -(cK,+)~~$
al) = PA =
of the statistical matrix
instead
are canonical
7~. 61. These
oscil-
quantities
state
operators
sin 2nE1.
of the system
the Wigner
is defined
de exp (-i(,P
L I P and Q are quantum
cos 2nli (5) by introducing
p. It turns out to be convenient
generalizing
this function
(4n)-sjd7
=
one can consider
PA, Q,I, which
-
by a density
of this operator
a nonrelativistic
P,
(4)
aAA)
1) :
(fit/2c~l)f (a;
where
+
aA, a;, two sets of real quantities
i(k~J2)f
ensemble
out explicitly,
or else the set of action
are interrelated
We
F.KA)(~A
mechanics.
of the amplitudes lator
-
phase-space
as follows
2,
distribution.
For
3) :
+ OQ)} Tr{p exp i(,>
associated
an
to use a
+ &}
to the classical
(6)
canonical
variables
Q. The
field
extension oscillators
of a, a+ (eq. expression expressed formed
to the system are concerned.
5) and
quantizing
here poses replacing
the latter
in the
no special j,
difficulty
Q by their
usual
way
as far as the
expression
immediately
in terms yields
an
for the Wigner function. The classical variables P, Q are then conveniently in terms of the action and angle variables and the result is Fourier-trans-
in the angles
E One obtains
/%2(V) =
7 stands
for the electromagnetic
a+ -
field:
eim+ Tr p exp .
(ex)-le_/%./~rJ-&
.{*d\/hr(ei” (Of course,
studied Indeed,
e-$+ a)}.
(7)
for the set 71, . . . . qn, . ...; rnt = c md~, etc). one needs a bolder generalization of the nonrelativistic
For the electrons-positrons
(rather than ordinary particles) concepts. The system is composed of N “chargeons” .nd these are endowed with spin. The natural generalization of the nonrelativistic formalism
3) consists
in writing
fW@(x,
p) = (&$-2N(N!)-l/dk
.Tr ,,: n /d,#++$)
ds eikr+isp.
e-ikz~‘-~r’~‘@+j):
(8)
i where
the
semicolons
denote
a normal
product.
Note
the
important
fact
that
the
ON THE STATISTICAL MECHANICS OF A RELATIVISTIC QUANTUM PLASMA
1601
Wigner function is now a matrix in spinor space. Going over to the momentum representation by using ( 1) and combining with (7) we get the expression of the Wigner function for the complete system:
Tr p(t) ++@(p -
@k)
exp[&&(ei+
a+ -
e--ig a)] +a(p + @k).
(9)
A particular case of this expression has been used by Rukhadze and Silin 4). After completion of this work we received the Ph. D. thesis of Irving 5) in which similar ideas are developed for a one-particle system without radiation. The equation of evolution of the Wigner function is obtained from the von Neuman equation atP = (@)[P. H3-
(10)
Combining with (9) and (4), one obtains, after evaluation of the commutators:
with
(14
(13) where
Equation (11) plays the role of the classical Liouville equation for this system. It is interesting to note that the “Liouville operator” is a finite displacement operator in the particle variables, and a first order differential operator in the field variables. Equation (11) stands in a form ready for the application of perturbation theory and diagram techniques 3) for its solution. The details of the present calculations, as well as further results will be published shortly.
1602
ON THE
STATISTICAL
We wish to thank acknowledge European
the Office,
Prof.
partial under
MECHANICS
I. Prigogine financial
contract
OF A RELATIVISTIC
for the interest
support nr. AF
of the
QUANTUM
he showed
Office
of
PLASMA
in this work.
Aerospace
We
Research,
61 (052)-179.
Received 3-8-65 R. BALESCU Faculti: des Sciences, Universite Libre de Bruxelles, Bruxelles, Belgique REFERENCES 11 Mangeney, A., Physica 29 (1963) 461. 2) Wigner, E., Phys. Rev. 40 (1932) 749. Moyal, J. E., Proc. Cambridge Phil. Sm. 45 (1949) 99. 3) Balescu, R., Statistical Mechanics of Charged Particles, Wiley-Interscience (1963). 4) Rukhadze, A. A. and Silin, V. P., J. exp. th. Ph. 39 (1960) 645. 5) Irving, J., Ph. D. Thesis, Princeton University (1965).
Publ. New York
Balescu, R. 1965
LETTER TO THE EDITOR On the statistical mechanics
of a relativistic
quantum plasma.
Although the problem deserves a great interest, it does not seem that the quantum statistical mechanics of relativistic systems has been developed in a systematic way, especially out of equilibrium. In particular the study of a relativistic electron-positron gas interacting with radiation is particularly interesting as it should lead to such important issues as the rigorous derivation of the laws of macroscopic electrodynamics, the hydrodynamics of a gas of Dirac particles, etc. We have started a program on this vast field, and we wish to sketch in the present letter the first results, giving the framework of the theory. We consider a system corresponding to a classical set of N electrons. As, however, the number of particles is not conserved in a relativistic system, the specification of the system has to be made by its charge, which is a well-defined concept. We therefore assume the system to be composed of N charge entities. This system can be realized by N, electrons and N, positrons, with N, - N, = N = const. The system is assumed to be imbedded in a neutralizing background of positive charge. The electromagnetic field produced by the charges is described by a set of transverse oscillators and by the electrostatic Coulomb interactions (Coulomb gauge). Let I@; t) be a solution of the Dirac one-particle equation; we expand it into a set of plane waves normalized in a cubic box of volume Q, and apply the second quantization, obtaining $a(%) = Q-a x { x [~a(&, S) b,,s + v*a(k; s) d~k,s]}.eik’r k s=fl (1) where ua(k, s)[va(k, s)] are plane-wave solutions of the Dirac equation, corresponding to wave vector k, spin s and positive [negative] energy; bks[dks] is the destruction operator for an electron [positron]. 4a(Rk) is defined by the quantity in curly brackets in the first line of eq. (1) : it is a 4-spinor consisting of a superposition of electron and positron states, playing the role of a “quasi particle”, which might be called a “chargeon”. This interpretation is supported by the fact that the well-known anticommutation relations for b and d imply the following relations: [P(k),
++@(k')l+ = 6aS&k,;
[W4, PWl+ The electromagnetic
= L++"(k), ++W')l+ = 0.
(4
vector potential is expanded in the usual way : (3)
where eL is a polarization unit vector, and I denotes the set of a wave vector KA and a polarization index ~11= 1, 2 (KA -6~ = 0 in Coulomb Gauge). The creation and
-
1599 -
1600
R. BALESCU
destruction
operators
a;,
an obey
the usual commutation
relations.
The hamiltonian
is then:
= x (RccP-k
H,
+ ,P’mc~) g+p(iik) ,$a(Rk)
k
H int =
Q-t I: I; (IIRc/K~)~ ~PY~AC#J +P(hk)
-e
kA P(ltk
The coulomb vistic
part is not written
statistical
If A(x) is regarded variables,
as a classical
as follows
now want
(a;
a description
characterized
representation
quantity
because
it is treated
system
fW(PQ)
(i.e. a c-number),
as in non-relati-
and angle
variables
+ aA) = QA =
(nc~,$+q$ -(cK,+)~~$
al) = PA =
of the statistical matrix
instead
are canonical
7~. 61. These
oscil-
quantities
state
operators
sin 2nE1.
of the system
the Wigner
is defined
de exp (-i(,P
L I P and Q are quantum
cos 2nli (5) by introducing
p. It turns out to be convenient
generalizing
this function
(4n)-sjd7
=
one can consider
PA, Q,I, which
-
by a density
of this operator
a nonrelativistic
P,
(4)
aAA)
1) :
(fit/2c~l)f (a;
where
+
aA, a;, two sets of real quantities
i(k~J2)f
ensemble
out explicitly,
or else the set of action
are interrelated
We
F.KA)(~A
mechanics.
of the amplitudes lator
-
phase-space
as follows
2,
distribution.
For
3) :
+ OQ)} Tr{p exp i(,>
associated
an
to use a
+ &}
to the classical
(6)
canonical
variables
Q. The
field
extension oscillators
of a, a+ (eq. expression expressed formed
to the system are concerned.
5) and
quantizing
here poses replacing
the latter
in the
no special j,
difficulty
Q by their
usual
way
as far as the
expression
immediately
in terms yields
an
for the Wigner function. The classical variables P, Q are then conveniently in terms of the action and angle variables and the result is Fourier-trans-
in the angles
E One obtains
/%2(V) =
7 stands
for the electromagnetic
a+ -
field:
eim+ Tr p exp .
(ex)-le_/%./~rJ-&
.{*d\/hr(ei” (Of course,
studied Indeed,
e-$+ a)}.
(7)
for the set 71, . . . . qn, . ...; rnt = c md~, etc). one needs a bolder generalization of the nonrelativistic
For the electrons-positrons
(rather than ordinary particles) concepts. The system is composed of N “chargeons” .nd these are endowed with spin. The natural generalization of the nonrelativistic formalism
3) consists
in writing
fW@(x,
p) = (&$-2N(N!)-l/dk
.Tr ,,: n /d,#++$)
ds eikr+isp.
e-ikz~‘-~r’~‘@+j):
(8)
i where
the
semicolons
denote
a normal
product.
Note
the
important
fact
that
the
ON THE STATISTICAL MECHANICS OF A RELATIVISTIC QUANTUM PLASMA
1601
Wigner function is now a matrix in spinor space. Going over to the momentum representation by using ( 1) and combining with (7) we get the expression of the Wigner function for the complete system:
Tr p(t) ++@(p -
@k)
exp[&&(ei+
a+ -
e--ig a)] +a(p + @k).
(9)
A particular case of this expression has been used by Rukhadze and Silin 4). After completion of this work we received the Ph. D. thesis of Irving 5) in which similar ideas are developed for a one-particle system without radiation. The equation of evolution of the Wigner function is obtained from the von Neuman equation atP = (@)[P. H3-
(10)
Combining with (9) and (4), one obtains, after evaluation of the commutators:
with
(14
(13) where
Equation (11) plays the role of the classical Liouville equation for this system. It is interesting to note that the “Liouville operator” is a finite displacement operator in the particle variables, and a first order differential operator in the field variables. Equation (11) stands in a form ready for the application of perturbation theory and diagram techniques 3) for its solution. The details of the present calculations, as well as further results will be published shortly.
1602
ON THE
STATISTICAL
We wish to thank acknowledge European
the Office,
Prof.
partial under
MECHANICS
I. Prigogine financial
contract
OF A RELATIVISTIC
for the interest
support nr. AF
of the
QUANTUM
he showed
Office
of
PLASMA
in this work.
Aerospace
We
Research,
61 (052)-179.
Received 3-8-65 R. BALESCU Faculti: des Sciences, Universite Libre de Bruxelles, Bruxelles, Belgique REFERENCES 11 Mangeney, A., Physica 29 (1963) 461. 2) Wigner, E., Phys. Rev. 40 (1932) 749. Moyal, J. E., Proc. Cambridge Phil. Sm. 45 (1949) 99. 3) Balescu, R., Statistical Mechanics of Charged Particles, Wiley-Interscience (1963). 4) Rukhadze, A. A. and Silin, V. P., J. exp. th. Ph. 39 (1960) 645. 5) Irving, J., Ph. D. Thesis, Princeton University (1965).
Publ. New York