Convenient expressions for commutators and anticommutators of phase space operators are obtained. The two-particle equilibrium distribution function is analyzed in terms of momentum-dependent quantum generalizations. In terms of f(rp, t), the ordinal number and moment density operators are given by.
These equations provide a first indication of the considerable formal similarity between f(rp, t) and the classical phase space density f (rp, t). As shown in Appendix A, the equal-time commutation relations of some first phase space operators can be expressed as. These operator equations are analogous to the equations of motion of the reduced Wigner distribution functions in the form originally given by Irving and Zwanzig.
Only the first term in the sine expansion survives the integration and gives The equations are equivalent to the slightly more cumbersome expressions for the quantum mechanical stress tensor and energy flux given by Puff and Gillis in terms of the $(r) and tlr+(r) operators.
DISTRIBUTION FUNCTIONS
The function n( 12) is best thought of as a special off-diagonal Fourier transform of the two-part density matrix. Diagrams la and lb contain a fully interacting one-particle propagator, and diagram lc means the sum of all two-particle coupled diagrams, so that Fig. Obviously, there are many functions besides N(kp) that reduce to n¢(p). ) in the classical limit; the reason for the special definitions (3. 10) and (3. 11) will become clear in section IV.
In addition to all the problems associated with the calculation of the quantum mechanical distribution function of pairs, one has to deal with the coupling of space and momentum variables. However, the first term in the perturbation expansion of H(kpp') is not difficult to obtain and serves to illustrate the general structure of h(kpp') and its relation to the classical h(k). the sum of the general two-part connected diagram with the impulse label in Fig. 3. In the limit h-0 the dependence on the impulse disappears and we restore the classical result of the first order, h (k) = -j3v( k).
CORRELATION FUNCTIONS
In particular, a calculation of the initial value of L would require a time or frequency analysis. To discuss the kinetic equation, it is convenient to use the complex argument function z defined by. In terms of F(kzpp'), the basic properties of the anticommutative correlation function can be summarized as.
However, the same result can be obtained more simply by exploiting the anticommutator property: the equilibrium average of Eq. A key property of F(kpp') is the fact that it has an inverse F-l(kpp') in the sense that. In addition to the static inverse F-1(kpp1), the derivation of the kinetic equation given in Section V makes essential use of the z-dependent inverse F-1(kzpp1), which satisfies.
Although there has been no clear proof, the existence of this z-dependent inverse seems well established in the classical. The existence of the static and z-dependent inverse is a special property of the anticommutative function and is not shared by any correlation function of interest. Therefore, the method used in the next section to derive a kinetic equation for F cannot be applied to X· In fact, it can be shown that X does not satisfy a kinetic equation of the same form as that for F.
QUANTUM KINETIC EQUATION
It should be noted here that a kinetic equation of the form (5.4) can be derived using a projection operator, as done in Appendix C. Since both methods lead to a kinetic equation of the form (5. 4) ), it turns out that using the modified-propagator expressions is equivalent to assuming the existence of the z-dependent inverse. Through the equation of motion (5. 4) the positivity and symmetry properties (4. 10) of F(kzpp') determine corresponding properties of the nucleus.
Conversely, if we obtain F(kzpp') as a solution of the kinetic equation (5.4) with an approximate kernel with these properties, then it will automatically satisfy the requirements of Eqs. In terms of K(kzpp'), the kernel properties are summarized as 5.11d) The static part is real and odd with respect to k- -k, but is itself not symmetric with respect to p-p'; instead, it is the sum of the static part and the flow term. There is an additional general property of the kernel that we can mention here, namely that the static kernel K(s)(kpp') is closely related to the connected part H(kpp') of the two-particle distribution function.
It should be noted that the initial condition used in this approach is the truncation of the exact initial condition. In the classical case, the approximation (5. 21) is equivalent to a direct truncation of L:(z), due to the special form of the classical F. This does not apply to the quantum 1: (z). Moreover, it should be noted that a direct truncation of the quantum mechanical I:{z) does not lead to a symmetric approximation.
In the first scheme, the initial condition and the static part of the kernel are given in terms of the truncated h {k); in the. It is clear that in the classical case the two schemes are equally feasible. For comparison, the large z-expansion of the exact F(kzpp} is given in terms of the frequency moments of S(kwpp') by.
If each factor in (5.27) could be evaluated exactly, instead of being truncated to second order as indicated, we would recover the expansion of the exact F(z), but, as given, the third and higher coefficients in the expansion of F(z) contains terms that do not appear in the expan-.
THE SECOND-ORDER KERNEL A. Static Part
It is equally simple to take the results for the two-link diagrams of F (12, 3) and substitute them into Eq. and H are the shapes of the general two-particle coupled diagram defined by Eqs. The expressions are written in terms of exact one- and two-particle distribution functions; truncated to second order, they give the full second-order expansion of the static kernel. The second-order term contains contributions from K(s) and . a part with two connected K~s) and K~s), and is given by.
We note that K{z) is odd in k, according to Eqs. 5. 11) and that its limit h-0 gives the correct second-order expansion of the classical result, Eq. To complete the list of static quantities appearing in the initial condition F(Z)(kpp'), we give the second-order term n(p), which is. This is not a circular definition because K~s) contains a first-order term in H. Up to this point we have been concerned with the diagrammatic analysis of simultaneous correlation functions, for which calculations are relatively straightforward.
It should be noted that there is no first-order contribution to the dynamic part. It should be noted that the statistical factor a(kp, k'p') decreases to its classical value (6. 18) in the limit of high temperature or low density,. We now proceed to transform the second-order kernel into such a form, starting by writing explicitly the shifts in p and p' indicated by the V operators.
Accordingly, the k,z-dependent nucleus involves collisions in the presence of a group of other particles, the collective effect of which is represented by a momentwn k and an energy w = Re z. From this expression it immediately follows that the nucleus of the second order satisfies the conditions for positivity and symmetry (5. 11). For example, the second-order dynamic kernel would have the same form as (6.21), but the function A.
In the long time, large distance limit the effect of these factors would disappear and we would again recover the Uehling-Uhlenbeck nucleus (6.25.
CONCLUSION
This will provide a kinetic equation applicable to a real quantum gas at densities for which a two-term virial expansion is the appropriate description of the equation of state. This requires attention to the order of the factors in subsequent expressions, but allows us to perform the integrations over k, k. A. 5), the gradient 'll act on both the functions to the right of it;. The evaluation of the diagrams displayed in the figures follows standard rules of many-body perturbation theory, with minor exceptions.
This appendix is not intended as a complete account of the method, but rather as a summary of the notation and special conventions used here. The basic object of the theory is assumed to be the imaginary-time-ordered n-particle momentum-space Green's function defined by . T is the time-ordered operator which rearranges the field operators from right to left in ascending order of their 1" arguments and inserts a factor of 1l1T, where 1T is the signature of .
This appendix contains an alternative derivation of the kinetic equation for F( 1, l'jz ), based on the use of a projection operator. We now insert a redundant factor Q on the left side of e -QiLtQ and move it to the other side of the anticomrnutator using . The differences are that the time dependence of the F-functions is determined by the modified propagator eQiLt, and that the last two factors in the second term of Eq.
In a practical evaluation of Eq. C. 12), one proceeds by re-expressing the correlation functions of the modified propagator in terms of the standard functions. We now turn to the question of the mathematical behavior of the expressions containing the modified propagator. Since no approximations were made, Eq. C. 7) is a formal identity; the projection operate r method therefore appears to provide an unambiguous derivation of an exact kinetic equation for F(l, l'Jz) of the form (5.
The Fourier transforms of the commutator and anticommutator functions are related by the fluctuation-distribution formula. Therefore, it is difficult to imagine that the expressions of the projection operator for the kernels in Eq. C. 19) are a useful starting point for physical approximations. This may be a warning that more subtle features of the n1ay projection operator also cause difficulties.
