Theoretical and Mathematical Physics, 143(3): 821–835 (2005)
A GENERALIZED COORDINATE–MOMENTUM REPRESENTATION IN QUANTUM MECHANICS L. S. Kuz’menkov∗ and S. G. Maksimov† We obtain a one-parameter family of (q, p)-representations of quantum mechanics; the Wigner distribution function and the distribution function we previously derived are particular cases in this family. We find the solutions of the evolution equations for the microscopic classical and quantum distribution functions in the form of integrals over paths in a phase space. We show that when varying canonical variables in the Green’s function of the quantum Liouville equation, we must use the total increment of the action functional in its path-integral representation, whereas in the Green’s function of the classical Liouville equation, the linear part of the increment is sufficient. A correspondence between the classical and quantum schemes holds only under a certain choice of the value of the distribution family parameter. This value corresponds to the distribution function previously found.
Keywords: (q, p)-representation, Liouville equation, path integral
1. Introduction According to the canonical quantization postulates, the Lie algebra of dynamical functions of the form A(q, p, t) (q, p ∈ Rs , where s is the number of degrees of freedom) is assigned the Lie algebra of Hermitian operators Aˆ in a Hilbert space such that the Poisson bracket { · , · } converts to the commutator (i/)[ · , · ]. But this assignment is nonunique because the canonically conjugate dynamical variables do not commute. As a rule, this ambiguity is removed by postulating an additional ordering procedure for the coordinate and momentum operators in the operator function A(ˆ q , pˆ, t) (the Weyl correspondence principle, the qˆpˆand pˆqˆ-quantizations, etc. [1]). The Weyl correspondence principle uniquely relates a Hermitian operator to a classical dynamical function using the Fourier integral: s s s s d u d v i(uˆq +vp) d q d p −i(uq+vp) W ˆ A(q, p) −→ A(ˆ q , pˆ) = e e A(q, p), (1) s (2π) (2π)s s where uq = i=1 ui qi . But this principle, like the qˆpˆ- and pˆqˆ-quantization rules, does not map the algebraic = (i/)[A, ˆ B] for the operation C = {A, B} for any dynamical functions A, B, and C to the operation C corresponding operators in L2 . Commutators of arbitrary operators also do not convert to the classical Poisson brackets. The commutator algebra of quantum mechanical operators in the Weyl representation is assigned not the Poisson bracket algebra of dynamical functions but the algebra of exotic Moyal brackets, which convert to the Poisson brackets in the semiclassical limit → 0: ← 2 −− → ← → i −− ∂p ∂q − ∂q ∂p B = {A, B} + O(2 ), {A, B}M = (A B − B A) = A sin (2) 2 ∗ Moscow
† Instituto
State University, Moscow, Russia, e-mail:
[email protected]. Tecnol´ ogico de Morelia, Morelia, Michoac´ an, M´ exico, e-mail:
[email protected].
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 401–416, June, 2005. Original article submitted November 22, 2004; revised January 20, 2005. c 2005 Springer Science+Business Media, Inc. 0040-5779/05/1433-0821
821
where A B is the Moyal product (or -product) [2], [3], ← → ← → −− −− ∂p ∂q − ∂q ∂p B. A B = A exp 2i No question about ordering the coordinate and momentum operators in operator functions can be posed such that relation (1) is no longer the definition of quantum operators based on the given classical dynamical functions A(q, p). This relation can be considered a particular representation of quantum mechanics where quantum mechanical equations of motion are most analogous to classical ones. In this case, operator commutators convert to Moyal brackets (2) for the corresponding “dynamical functions.” Such a function ˆ can be expressed of coordinates and momenta A(q, p) in (1) (or the Wigner representation of the operator A) in terms of its kernel: ˆ + ξ/2 eipξ/ . AW (q, p) = ds ξ q − (ξ/2)|A|q (3) It is real if the corresponding operator Aˆ is Hermitian. In the Wigner representation, the quantum mean of the operator Aˆ can be expressed as the statistical mean with the Wigner function fW (q, p): ˆ = tr(ˆ ˆ =
A ρA) fW (q, p, t) =
ds q
1 (2π)s
ds p AW (q, p)fW (q, p),
ds ξ q − (ξ/2) ρˆ(t) q + ξ/2 eipξ/ ,
where ρˆ is the von Neumann density matrix [4]. The evolution of the Wigner function is described by the equation ∂t fW + {HW , fW }M = 0, which converts to the classical Liouville equation in the limit → 0. Here, the Hamiltonian function HW is also the Wigner representation of the Hamiltonian operator H. We considered an alternative way to introduce a distribution function in quantum mechanics in [5]–[8] based on directly quantizing the classical microscopic distribution function that is a product of the density of the probability of finding the system at the point p ∈ Rs of the momentum space and the density of the probability of finding the system at the point q ∈ Rs of the coordinate space: f (q, p, t) = n(p, t)n(q, t),
(4)
where n(q, t) =
s δ qi − q i (t) ≡ δ q − q(t) = i=1
n(p, t) =
s δ pi − pi (t) ≡ δ p − p(t) = i=1
ds v iv(p−q(t))/ e , (2π)s ds u iu(q−p(t))/ e , (2π)s
and q(t) and p(t) are canonical variables. The corresponding quantum operator distribution function is a product of the operators n ˆ (p) and n ˆ (q): fˆ(q, p) = 822
ds v iv(p−p)/ ˆ e (2π)s
ds u iu(q−ˆq )/ e . (2π)s
(5)
Here, the operator fˆ(q, p) is not Hermitian, and the corresponding distribution function f (q, p, t) = tr fˆ(q, p)ˆ ρ(t) is complex. The kernel of operator (5) is given by
q |fˆ(q, p)|q = q |p p|q q|q ,
(6)
while the quantum mean of this operator is f (q, p, t) =
ρ(t)|q = p|q q|ˆ ρ(t)|p . ds q ds q q |p p|q q|q q |ˆ
(7)
The Wigner function fW (q, p, t) and the function f (q, p, t) are “normalized” like the classical distribution function, s s (8) d q d p fW (q, p, t) = ds q ds p f (q, p, t) = 1 (strictly speaking, the integrals in (8) cannot serve as definitions of norms of the functions fW and f ), and have a correct classical limit as → 0. At the same time, the function fW (q, p, t) is not positive definite, while f (q, p, t), which is meant to specify a state of the system along with the wave function, is not real. We can construct other distribution functions that differ from the functions considered above. We see that we cannot define the principles for selecting the most preferred distributions without solving the problem of finding a family of distribution functions to which the functions fW (q, p, t) and f (q, p, t) belong, without deducing and solving the evolution equations for this family, and without analyzing the solutions. In this paper, we deduce a generalized form of different (q, p)-representations of quantum mechanics dependent on an arbitrary parameter χ ∈ R. The Wigner distribution function fW (q, p, t) and the function f (q, p, t) given by (7) appear to be particular cases corresponding to the respective parameter values χ = 1/2 and χ = 0. The evolution of the distribution function fχ (q, p, t) in the generalized (q, p)-representation is found in the form of a path integral. We show that when varying the canonical variables q and p in the classical Green’s function, the linear part of the increment of the action functional is sufficient, whereas in the quantum Green’s function, we must use its total increment. There exists a unique (q, p)-representation of quantum mechanics where the action functionals in the Green’s function of the quantum Liouville equation correspond to the classical variational scheme. This representation corresponds to the parameter value χ = 0. Another possible representation corresponding to χ = 1 is dual to the representation χ = 0 and is therefore not considered independent.
2. A generalized (q, p)-representation We can deduce the most general form of the “Wigner-like” distribution functions based on the following considerations. The classical microscopic distribution function for a dynamical system with s degrees of freedom is the product of probability densities (4). On one hand, function (4) is a dynamical function because it depends on the particle coordinates q¯(t) and particle momenta p¯(t); on the other hand, it is a field function of the variables q and p. We can represent formula (4) as the Fourier integral ds v iv(p−p(t)) ds u iu(q−¯q (t)) ¯ f (q, p, t) = e e = (2π)s (2π)s s s d ud v = exp iαv p − p¯(t) × 2s (2π) × exp i (1 − α)v p − p¯(t) + (1 − β)u q − q¯(t) exp iβu q − q¯(t) , (9) 823
where α and β are arbitrary numbers. We consider a quantum generalization of formula (9) in which q¯(t) and p¯(t) in the right-hand side are replaced with the corresponding operators, fˆ(q, p) =
ds u ds v exp iαv(p − pˆ)/ × 2s (2π)
× exp i (1 − α)v(p − pˆ) + (1 − β)u(q − qˆ) / exp iβu(q − qˆ)/ .
(10)
Because the relation
exp i(1 − α)v(p − pˆ)/ + i(1 − β)u(q − qˆ)/ =
= exp −i(1 − α)(1 − β)vu/2 exp i(1 − α)v(p − pˆ)/ exp i(1 − β)u(q − qˆ)/ holds by the Weyl identity, formula (10) yields an operator generalization of the distribution function in an arbitrary (q, p)-representation: fˆχ (q, p) =
ds u ds v −iχvu/ iv(p−p)/ ˆ e e eiu(q−ˆq )/ , (2π)2s
(11)
where χ = (1 − α)(1 − β)/2. The requirement that operator (11) be Hermitian (fˆχ+ = fˆχ ) immediately leads to the value χ = 1/2, and we obtain Wigner function fˆχ=1/2 = fˆW . The parameter value χ = 0 corresponds to distribution function (5), fˆχ=0 = fˆ. We now find the kernel of operator fˆχ in the generalized (q, p)-representation. We have
q |fˆχ (q, p)|q =
=
ds u ds v −iχvu/ e (2π)2s ds u ds v −iχvu/ e (2π)2s
1 = e−ip(q −q )/ (2π)s
ˆ |p p |eiu(q−ˆq )/ |q = ds p q |eiv(p−p)/
ds p eiv(p−p )/ eiu(q−q
)/
q |p p |q =
ds u exp iu q − χq − (1 − χ)q / . (2π)s
(12)
The integral in the right-hand side of (12) diverges for Im χ = 0. We should therefore take Im χ = 0. We then obtain 1
q |fˆχ (q, p)|q = (13) e−ip(q −q )/ δ q − χq − (1 − χ)q s (2π) from Eq. (12), and evaluating the quantum mean of such an operator, we find the quantum distribution function in the generalized (q, p)-representation: fχ (q, p, t) = tr fˆχ (q, p)ˆ ρ(t) = s s d q d q −ip(q −q )/ = e δ q − χq − (1 − χ)q q |ˆ ρ|q = s (2π) ds ξ ipξ/ = q − χξ|ˆ ρ|q + (1 − χ)ξ . e (2π)s
824
(14)
3. Evolution of the classical distribution function in the path-integral form The Liouville equation for microscopic distribution function (4) of a system with s degrees of freedom and a Hamiltonian H(q, p, t) directly follows from the definition of this function and the Hamilton equations and can be represented as the Schr¨odinger equation in the phase space, ∂t f = [H]f,
ft=t0 = f0 ,
(15)
where [H] is the differential operator s ∂H ∂ ∂H ∂ ∂H ∂ ∂H ∂ − [H] = − ≡ ∂qi ∂pi ∂pi ∂qi ∂q ∂p ∂p ∂q i=1
(16)
and f (q, p, t0 ) = f0 (q, p) is an initial distribution. In the framework of such an interpretation of the Liouville equation, it is convenient to treat the Fourier expansion of the state vector f (q, p) with respect to some complete orthogonal set of functions as a transition from one representation of some abstract state vector |f ) to another. We consider the function f (q, p) the (q, p)-representation of this vector: f (q, p) = (q, p|f ). The transition to another (Q, P )-representation (the Fourier transformation) is realized by f (Q, P ) = (Q, P |f ) =
ds q ds p (Q, P |q, p)(q, p|f ).
The orthonormality and completeness of the transition functions (q, p|Q, P ) are expressed by ds q ds p (Q, P |q, p)(q, p|Q , P ) = (Q, P |Q , P ) = δ(Q − Q )δ(P − P ),
ds Q ds P (q, p|Q, P )(Q, P |q , p ) = (q, p|q , p ) = δ(q − q )δ(p − p ).
The normalization condition then changes to
ds q ds p q, p|f (t) =
µ(Q, P )f (Q, P, t)ds Q ds P = 1,
(17)
where µ(Q, P ) = ds q ds p (q, p|Q, P ). We intend to obtain the solution of the initial problem for the microscopic distribution function. For this, we find the kernel of the operator [H] in the (q, p)-representation: [H]f (q, p) ≡ (q, p|[H]|f ) = ds q ds p (q, p|[H]|q , p )(q , p |f ) =
∂H ∂ ∂H ∂ − f (q, p) = ∂q ∂p ∂p ∂q ∂H ∂ ∂H ∂ s s − = d q d p δ(q − q )δ(p − p )f (q , p ). ∂q ∂p ∂p ∂q =
We thus obtain ∂H(q, p) ∂δ(p − p ) ∂H(q, p) ∂δ(q − q ) q, p|[H]|q , p = δ(q − q ) − δ(p − p ) . ∂q ∂p ∂p ∂q
(18) 825
The solution of problem (15) is q, p|f (t) = q, p|G(t, t0 )|f (t0 ) = = ds q0 ds p0 G(q, p, t|q0 , p0 , t0 ) q0 , p0 |f (t0 ) ,
(19)
where the propagator G(t, t0 ) is the operator of the form t G(t, t0 ) = exp H(τ ) dτ ,
(20)
t0
which belongs to a one-parameter Lie group. Dividing the time segment [t0 , t] into n segments [τk , τk+1 ], where k = 0, (n − 1), τ0 = t0 , and τn = t, we find G(t, t0 ) =
n
G(τk , τk−1 ).
k=1
Using the completeness of the set of vectors |q, p), we obtain the equation
G(q, p, t|q0 , p0 , t0 ) =
n−1
ds qj ds pj
j=1
n
G(qk , pk , τk |qk−1 , pk−1 , τk−1 )
(21)
k=1
for the kernel of the operator G(t, t0 ), where qn = q and pn = p. Taking formulas (18) and (20) into account, we have
τk
G(qk , pk , τk |qk−1 , pk−1 , τk−1 ) = qk , pk exp H(τ ) dτ qk−1 , pk−1 τk−1
δ(qk − qk−1 )δ(pk − pk−1 ) + (qk , pk | H(τk ) |qk−1 , pk−1 )∆τk = ∂H(qk , pk , τk ) ∂δ(pk − pk−1 ) = δ(qk − qk−1 )δ(pk − pk−1 ) + δ(qk − qk−1 ) − ∂qk ∂pk ∂H(qk , pk , τk ) ∂δ(qk − qk−1 ) ∆τk , − δ(pk − pk−1 ) ∂pk ∂qk where ∆τk = τk − τk−1 . Because δ(qk − qk−1 ) =
ds uk iuk (qk −qk−1 ) e , (2π)s
we find
δ(pk − pk−1 ) =
ds vk −ivk (qk −qk−1 ) e , (2π)s
ds uk ds vk exp i(uk (qk − qk−1 ) − vk (pk − pk−1 )) × 2s (2π) ∂H(qk , pk , τk ) ∂H(qk , pk , τk ) × 1 − i vk + uk ∆τk ∂qk ∂pk s ∂H(qk , pk , τk ) d uk ds vk exp iuk qk − qk−1 − ∆τk − (2π)2s ∂pk ∂H(qk , pk , τk ) ∆τk − ivk pk − pk−1 + . ∂qk
G(qk , pk , τk |qk−1 , pk−1 , τk−1 )
826
(22)
It remains to substitute formula (22) in Eq. (21) and pass to the limit max{∆τk } → 0. As a result, we obtain the Green’s function of the Liouville equation in the path-integral form:
G(q, p, t|q0 , p0 , t0 ) =
1 max{∆τk }→0 (2π)s lim
n−1 n ds qk ds pk ds uk ds vk × (2π)s (2π)s k=1
k=1
n ∂H(qk , pk , τk ) ∂H(qk , pk , τk ) × exp iuk q˙k − − ivk p˙k + ∆τk ≡ ∂pk ∂qk k=1
≡
1 (2π)s
s d q(τ ) ds p(τ ) ds u(τ ) ds v(τ ) τ
(2π)s
(2π)s
τ
×
t t ∂H(q, p, τ ) ∂H(q, p, τ ) × exp i u q˙ − v −p˙ − dτ + i dτ , ∂p ∂q t0 t0
(23)
where q˙k =
lim
max{∆τk }→0
qk − qk−1 , ∆τk
p˙ k =
lim
max{∆τk }→0
pk − pk−1 . ∆τk
A formal analogy between the classical and quantum descriptions of the evolution of a system has long been known. Bogoliubov and Bogolyubov Jr. indicated in [9] that “it can be used to extend some methods of quantum mechanics to the case of studying the evolution. . . of classical systems.” In fact, we have used this recommendation here, which allows comparing to establish which distribution function in the family under consideration provides the most complete analogy between the classical and quantum microscopic descriptions of the evolution of the same system of interacting particles.
4. A distribution function in quantum mechanics and the generalized (q, p)-representation As is known, a quantum analogue of the classical Liouville equation is the equation for the density matrix ρˆ: ∂t ρˆ =
1 [H, ρˆ], i
ρˆ(t0 ) = ρˆ0 .
(24)
The solution of problem (24) is given by the unitary transformation, (t, t0 )ˆ −1 (t, t0 ), ρˆ(t) = U ρ(t0 )U where
i t U(t, t0 ) = T exp − dτ H(τ ) , t0
(25)
−1 (t, t0 ) = U (t0 , t), U
and hence (t, t0 )ρ(t0 )U −1 (t, t0 ) = tr ρˆ(t0 ) = 1. tr ρˆ(t) = tr U The existing analogy between the density matrix and classical state vector allows assuming that the Hermitian operator ρˆ can differ from the state ket vector by a coefficient ζ > 0: |f ) = ζ ρˆ.
(26) 827
Let |A) be an arbitrary vector of the space of dynamical functions. Then its observable A is a functional of the form ˆ
A = (f |A) = tr(ζ ρˆζ −1 A).
(27)
ˆ It follows from (27) that |A) = ζ −1 A. The matrix elements of the density matrix operator define the specific representation of the state vector |f ): (28)
q |ζ ρˆ|q = tr |q q |ζ ρˆ = (q , q |f ). Along with the bra vectors (q , q | = |q q |, we can consider the ket vectors |q , q ) = |q q | = (q , q |+ .
The set |q , q ) forms an orthogonal system of vectors: q , q˜ ) = tr |q q |˜ q ˜ q | = δ(q − q˜ )δ(q − q˜ ). (q , q |˜
The completeness of |q , q ) follows from the completeness of the system of the eigenvectors |q of the position operator, and we therefore have |A) =
s
s
d q d q |q , q )(q , q |A) ≡
ˆ ds q ds q |q q | q |ζ −1 A|q
for an arbitrary vector. The dynamical function A(q, p) in the phase space is the (q, p)-representation of the abstract vector |A). We must pass from the (q , q )-representation to the (q, p)-representation using a linear functional: A(q, p) ≡ (q, p|A) =
ˆ . ds q ds q (q, p|q , q ) q |ζ −1 A|q
(29)
Specifying the transition function (q, p|q , q ) completely determines the form of a dynamical function, including the distribution function f (q, p). With (29) taken into account, the quantum mean of the operator Aˆ in the new (q, p)-representation takes the form of a statistical mean:
A = =
ds q ds p f ∗ (q, p)A(q, p) = ds q ds q q |ζ ρˆ|q
ds q ds p (q , q |q, p)A(q, p),
whence it follows with (27) taken into account that
ˆ , ds q ds p (q , q |q, p)A(q, p) = q |ζ −1 A|q
(30)
where (q , q |q, p) = (q, p|q , q )∗ . The orthogonality and completeness of the system of vectors (q , q |q, p) follows from Eqs. (29) and (30):
828
q , p˜) = δ(q − q˜)δ(p − p˜), ds q ds q (q, p|q , q )(q , q |˜
(31)
ds q ds p (q , q |q, p)(q, p|˜ q , q˜ ) = δ(q − q˜ )δ(q − q˜ ).
(32)
Based on formula (13), we can construct the functions with properties (31) and (32): (q, p, χ|q , q ) = (2π)s/2 q |fˆχ (q, p)|q = 1 e−ip(q −q )/ δ q − χq − (1 − χ)q . (33) s/2 (2π)
Taking into account that the set of functions (q, p, χ|q , q ) forms a complete orthogonal system and comparing Eqs. (14) and (29), we obtain |q, p) = |q, p, χ) and ζ = (2π)−s/2 , i.e., =
|f ) = (2π)−s/2 ρˆ,
ˆ |A) = (2π)s/2 A.
The evolution of the distribution function f (q, p, t), like the evolution of the classical state vector and the dynamics of the density matrix, must have the form of the transformation f (q, p, t) = ds q0 ds p0 GQ (q, p, t|q0 , p0 , t0 )f (q0 , p0 , t0 ), (34) where the propagator GQ (q, p, t|q0 , p0 , t0 ) is an element of a one-parameter Lie group. Taking map (29) and its inverse into account and using Eq. (25) for the dynamics of the density matrix, we find f (q, p, t) = ds q ds q (q, p|q , q ) × × =
t0 )|q0 q0 |ζ ρˆ(t0 )|q0 q0 |U (t0 , t)|q = ds q0 ds q0 q |U(t, s
s
d q d q (q, p|q , q ) ×
t0 )|q0 q0 |U (t0 , t)|q × ds q0 ds q0 q |U(t,
ds q0 ds p0 (q0 , q0 |q0 , p0 )f (q0 , p0 , t0 ).
Comparing this equation with Eq. (34), we obtain the explicit form of the propagator in the (q, p) (t, t0 ) and the representation. It is expressed in terms of the matrix elements of the unitary operator U transition functions (q, p|q , q ) and (q , q |q, p): s s (t, t0 )|q0 × GQ (q, p, t|q0 , p0 , t0 ) = d q d q ds q0 ds q0 q |U (t0 , t)|q (q, p|q , q )(q0 , q0 |q0 , p0 ). × q0 |U
(35)
We consider the evolution of a dynamical system within a small time interval ∆t. Taking t0 = t − ∆t, we obtain GQ (q, p, t|q0 , p0 , t − ∆t) = ds q ds q ×
i t
)
q × ds q0 ds q0 q T exp − dτ H(τ 0 t−∆t t
i
)
q (q, p|q , q )(q0 , q0 |q0 , p0 ) = × q0 T exp dτ H(τ t−∆t
×
= δ(q − q0 )δ(p − p0 ) + ∆t s s + ds q0 ds q0 (q, p|q , q )(q0 , q0 |q0 , p0 ) × d q d q i 2 × q |H(t)|q 0 δ(q0 − q ) − δ(q − q0 ) q0 |H(t)|q + O (∆t)
(36) 829
from (35). On the other hand, GQ (q, p, t|q0 , p0 , t − ∆t) =
q, p
exp
t
t−∆t
H(τ ) Q dτ
q0 , p0 =
= δ(q − q0 )δ(p − p0 ) + ∆t(q, p| H(t) Q |q0 , p0 ) + O (∆t)2 ,
(37)
where H(t) Q is the quantum generalization of classical operator (16). Using Eqs. (36) and (37), we obtain the kernel of the operator [H]Q : 1 (q, p|[H]Q |q0 , p0 ) = i
s
s
d q d q
ds q0 ds q0 (q, p|q , q )(q0 , q0 |q0 , p0 ) ×
0 δ(q0 − q ) − δ(q − q0 ) q0 |H|q . × q |H|q
(38)
Treating the generalized (q, p)-representation as a (q, p)-representation, i.e., taking |q, p) = |q, p, χ), we obtain 1 q, p|[H]χ |q0 , p0 ) = i ×
s
s
d q d q
ds q0 ds q0 ×
exp{−(i/)p(q − q )} δ q − χq − (1 − χ)q × s/2 (2π)
exp{(i/)p0 (q0 − q0 )} δ q0 − χq0 − (1 − χ)q0 × s/2 (2π) 0 δ(q0 − q ) − δ(q − q0 ) q0 |H|q × (q, p|q , q )(q0 , q0 |q0 , p0 ) q |H|q
×
from (38). Using the change of variables q0 = q˜0 + (1 − χ)ξ0 , q0 = q˜0 − χξ0 , q = q˜ + (1 − χ)ξ, and q = q˜ − χξ, we bring the matrix elements of the operator [H]χ to the form 1 (q, p|[H]χ |q0 , p0 ) = i ×
i ds ξ ds ξ0 (pξ − p0 ξ0 ) × exp (2π)s 0 − χξ0 δ q0 − q + (1 − χ)(ξ0 − ξ) − q − χξ|H|q
+ (1 − χ)ξ . − δ q − q0 − χ(ξ − ξ0 ) q0 + (1 − χ)ξ0 |H|q
(39)
in (39) in terms of the dynamical Using Eq. (30), we can express the matrix elements of the operator H function H(q, p): i ds Q ds P P q − q exp − χ(ξ − ξ ) Hχ (Q, P ) × 0 0 (2π)s × δ Q − χq0 + χ2 ξ0 − (1 − χ)q + χ(1 − χ)ξ , (40) s i d Q ds P + (1 − χ)ξ = q0 + (1 − χ)ξ0 |H|q P q exp − q + (1 − χ)(ξ − ξ) Hχ (Q, P ) × 0 0 (2π)s × δ Q − χq − χ(1 − χ)ξ − (1 − χ)q0 − (1 − χ)2 ξ0 . (41) 0 − χξ0 = q − χξ|H|q
830
Substituting (40) and (41) in Eq. (39), we find s i ds ξ ds ξ0 d Q ds P (pξ − p0 ξ0 ) × Hχ (Q, P ) exp (2π)s (2π)s i × exp P q − q0 − χ(ξ − ξ0 ) δ Q − χq0 + χ2 ξ0 − (1 − χ)q + χ(1 − χ)ξ × × δ q0 − q + (1 − χ)(ξ0 − ξ) − s s s i d Q ds P 1 d ξ d ξ0 H (Q, P ) exp ξ ) × − (pξ − p χ 0 0 i (2π)s (2π)s i P q0 − q + (1 − χ)(ξ0 − ξ) δ q − q0 − χ(ξ − ξ0 ) × × exp (42) × δ Q − χq − χ(1 − χ)ξ − (1 − χ)q0 − (1 − χ)2 ξ0 .
1 (q, p|[H]χ |q0 , p0 ) = i
In the first term in the right-hand side of Eq. (42), we pass from integrating over the variables Q and P to integrating over the variables v and u related to Q and P by Q = q + χv and P = p + (1 − χ)u. In the second term in the right-hand side of (42), we change the variables as Q = q − (1 − χ)v and P = p − χu. We then obtain (q, p|[H]χ |q0 , p0 ) = × = ×
ds u ds v exp iu(q − q0 ) − iv(p − p0 ) × 2s (2π) 1 Hχ q + χv, p + (1 − χ)u − Hχ q − (1 − χ)v, p − χu = i
ds u ds v exp iu(q − q0 ) − iv(p − p0 ) × (2π)2s
1 exp χv∂q + (1 − χ)u∂p − exp −(1 − χ)v∂q − χu∂p Hχ (q, p) i
(43)
from (42). Formulas (38) and (43) also hold for the operators of arbitrary dynamical functions. In particular, (43) remains valid under changing the function Hχ to Aχ and the operator [H]χ to [A]χ in the respective right- and left-hand sides.
5. Evolution of the density matrix in the generalized (q, p)-representation We find the propagator Gχ (q, p, t|q0 , p0 , t0 ) in the path-integral form [1], [10]. Taking the group property of the propagator t H(τ ) χ dτ Gχ (t, t0 ) = exp t0
and the orthogonality and completeness of the system of the vectors |q, p, χ) given by (33) into account and dividing the time segment [t0 , t] into n segments, we obtain
Gχ (q, p, t|q0 , p0 , t0 ) =
n−1 j=1
ds qj ds pj
n
Gχ (qk , pk , τk |qk−1 , pk−1 , τk−1 ),
(44)
k=1
831
where qn = q and pn = p. Now taking formula (43) into account, we find Gχ (qk , pk , τk |qk−1 , pk−1 , τk−1 ) (qk , pk |1 + H(τk ) χ ∆τk |qk−1 , pk−1 ) =
ds uk ds vk exp i uk (qk − qk−1 ) − vk (pk − pk−1 ) × 2s (2π) × 1 − i−1 Hχ qk + χvk , pk + (1 − χ)uk , τk − − Hχ qk − (1 − χ)vk , pk − χuk , τk ∆τk s d uk ds vk exp i uk (qk − qk−1 ) − vk (pk − pk−1 ) − (2π)2s − −1 Hχ qk + χvk , pk + (1 − χ)uk , τk − − Hχ qk − (1 − χ)vk , pk − χuk , τk ∆τk , =
where ∆τk = τk − τk−1 . Substituting this result in formula (44) and passing to the limit as max{∆τk } → 0, we obtain the formula for the propagator Gχ in the path-integral form: 1 Gχ (q, p, t|q0 , p0 , t0 ) = lim max{∆τk }→0 (2π)s
n−1 n ds qj ds pj ds uk ds vk × (2π)s (2π)s j=1 k=1
n uk (qk − qk−1 ) − vk (pk − pk−1 ) − × exp i k=1
− −1 Hχ qk + χvk , pk + (1 − χ)uk , τk − − Hχ qk − (1 − χ)vk , pk − χuk , τk ∆τk ≡ ≡
1 (2π)s
t s d q(τ ) ds p(τ ) ds u(τ ) ds v(τ ) exp i dτ uq˙ − v p˙ − s s (2π) (2π) t0 τ τ
− −1 Hχ q + χv, p + (1 − χ)u, τ − Hχ q − (1 − χ)v, p − χu, τ ,
(45)
which must be supplemented with the boundary conditions q(t) = q, p(t) = p and q(t0 ) = q0 , p(t0 ) = p0 . We pass to the classical limit by expanding the Hamiltonian function in (45) in a power series in the Planck constant: s d q(τ ) ds p(τ ) ds u(τ ) ds v(τ ) 1 Gχ (q, p, t|q0 , p0 , t0 ) = × (2π)s (2π)s (2π)s τ τ t t ∂Hχ (q, p, τ ) ∂Hχ (q, p, τ ) × exp i dτ u q˙ − dτ v −p˙ − +i + O() . ∂p ∂q t0 t0 Green’s function (45) of the quantum Liouville equation thus converts to formula (23) for the classical propagator in the limit → 0. Setting u = δp/ and v = δq/ in (45) and taking the boundary conditions δq(t0 ) = δp(t0 ) = 0 and t δq(t) = δq, δp(t) = δp into account, we can represent the integral i t0 dτ (uq˙ − v p) ˙ as i 832
t
t0
t
i i t dτ (δpq˙ − δq p) ˙ = − δqp
+ dτ p + (1 − χ)δp (q˙ + χδ q) ˙ + (p − χδp) q˙ − (1 − χ)δ q˙ . t0 t0
The Green’s function of the quantum Liouville equation in the generalized (q, p)-representation then becomes Gχ (Ω, t|Ω0 , t0 ) =
1 (2π)s
dΩ(τ ) dδΩ(τ ) × s (2π) (2π)s τ τ
i i × exp − δqp + Att0 q + χδq, p + (1 − χ)δp − t − At0 q − (1 − χ)δq, p − χδp ,
(46)
where Att0 [q, p] is the action functional of the classical form. Similarly changing the variables in (23) as u = δp/ and v = δq/, we obtain the Green’s function of the classical Liouville equation in the form 1 G(Ω, t|Ω0 , t0 ) = (2π)s
i t dΩ(τ ) dδΩ(τ ) i exp − δqp + δAt0 [q, p] . (2π)s (2π)s τ τ
(47)
We see that when varying the canonical variables q and p in the classical Green’s function, we can restrict ourself to the linear part of the increment of the action functional, whereas in the quantum Green’s function with an arbitrary value of the parameter χ, we must use the total increment of this functional. The variations δq(t0 ) and δp(t0 ) are equal to zero only at the initial instant. In the quantum Green’s function, the values of both the coordinate and momentum at the final instant t are varied in the functionals Att0 q + χδq, p + (1 − χ)δp and Att0 q − (1 − χ)δq, p − χδp , whereas in the classical scheme, it is assumed that either the variations of the initial and final coordinates are equal to zero (δq(t0 ) = δq(t) = 0) for arbitrary values of momenta or the variations of the initial and final momenta are equal to zero (δp(t0 ) = δp(t) = 0) for nonfixed coordinates. Such a disagreement between the variational methods of evaluating the Green’s function for an arbitrary χ ∈ R can be removed by a proper choice of this parameter. We must take either χ = 0 or χ = 1 in formula (46). In particular, we set p + u = p and q − v = q in the case χ = 0. We then find Gχ=0 (Ω, t|Ω0 , t0 ) =
ds qn ds pn (2π)2s
s d q(τ ) ds p (τ ) ds q (τ ) ds p(τ ) × (2π)s (2π)s τ τ
i i t t ˜ × exp − (pq − p0 q0 ) + At0 [q, p ] − At0 [q , p] ,
(48)
where Att0 [q, p ] is the action of the classical type with the respective fixed initial and final coordinates q(t0 ) = q0 and q(t) = q, t dτ p q˙ − Hχ=0 (q, p , τ ) , Att0 [q, p ] = t0
while A˜tt0 [q , p] is the action of the classical type with the respective fixed initial and final momenta p(t0 ) = p0 and p(t) = p, t t ˜ At0 [q , p] = dτ −q p˙ − Hχ=0 (q , p, τ ) . t0
In this case, the transition function in the (q, p)-representation is
(q , q |q, p, 0) =
eip(q −q )/ δ(q − q ), (2π)s/2 833
and we obtain a compact formula for the vector |q, p, 0): |q, p, 0) = (2π)s/2 |q q|p p|.
(49)
In the case χ = 1, the vector |q, p, 1) is |q, p, 1) = (2π)s/2 |p p|q q| = |q, p, 0)+ = (q, p, 0|.
(50)
Therefore, the value χ = 1 does not lead to qualitatively new results; the |q, p, 1)-representation proves to be dual to representation (49) of quantum mechanics, and the Green’s function Gχ in the |q, p, 1)-representation is related to function (48) by complex conjugation. The dynamical functions in the considered (q, p)-representation with χ = 0 are Aχ=0 (q, p) = (q, p, 0|A) = ˆ whence it directly follows that (2π)s p|q q|A|p , ˆ = Aχ=0 (q, p) q|p .
q|A|p
(51)
Formula (51) coincides in form with the formula for the kernel of the operator in the so-called qˆpˆ-quantization where all operators in L2 are considered polynomials of the form [1] Aˆ =
m
m
Am1 ,...,mn ;m1 ,...,mn qˆ1m1 · · · qˆnmn pˆ1 1 · · · pˆn n
(52)
corresponding to the classical dynamical functions
A(q, p) =
m
m
Am1 ,...,mn ;m1 ,...,mn q1m1 · · · qnmn p1 1 · · · pn n .
ˆ = A(q, p) q|p . The fundamental difference of representation (51) It easily follows from Eq. (52) that q|A|p from the qˆpˆ-quantization is that (51) is a definition of a quantum dynamical function Aχ=0 (q, p) given a known operator Aˆ in L2 with no additional assumptions on its form. Formula (52) is destined for finding the form of the operator function A(ˆ q , pˆ) based on the classical dynamical function A(q, p) satisfying the algebra of the classical Poisson brackets. The set of functions Aχ=0 , to which the quantum distribution function also belongs, does not coincide with the Lie algebra based on the classical Poisson brackets.
6. Conclusion We have obtained the generalized (q, p)-representation of quantum mechanics, including its special representation (χ = 0) in which the variational methods are universal for the quantum and classical schemes. The properties of quantum dynamical systems in the (q, p)-representation with χ = 0 and also the results obtained for the Green’s function of the quantum Liouville equation as a functional on the solutions of the classical equations of motion δAtt0 /δΩ = 0 and δ A˜tt0 /δΩ = 0 in the canonical form is the content of the following paper. Acknowledgments. This work was supported in part (S. G. M.) by the Consejo Nacional de Ciencia y Tecnolog´ıa de M´exico (CONACYT). 834
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