Appendix F Fokker–Planck Equations
Appendix F Fokker–Planck Equations
(p.360) F.1 Correspondence Rules
F.1.1 Grassmann and Operator Formulae
In the Liouville–von Neumann equation, the density operator may be replaced by or or , where and are boson or fermion operators. There will be a consequent change in the characteristic function and hence in the distribution function. These changes are expressed as correspondence rules.
The definition of the characteristic function is
where and are two sets of c-numbers associated with the p bosonic modes, and are two sets of Grassmann numbers associated with the n fermionic modes.
The characteristic function and the distribution function are related via Grassmann and c-number integrals:
where and are also two sets of c-numbers associated with the p bosonic modes, and and are also two sets of Grassmann numbers associated with the n fermionic modes.
The correspondence rules for fermion annihilation and creation operators may be obtained by starting with the canonical form of the density operator for the combined (p.361) boson–fermion system. There will be a consequent change in the Bargmann state projectors and hence in the distribution function. These changes are expressed as correspondence rules.
The density operator is related to the canonical form of the distribution function as follows:
is a product of normalised projectors, with and being the usual fermion and boson Bargmann state projectors:
The two forms for the density operator apply because is an even Grassmann operator and therefore commutes with .
(p.362) The derivation of the correspondence rules involves using standard quantum rules such as
involving operators and to change the order of exponentials and other operators for the boson terms. In addition, the commutation of boson operators with fermion operators is used extensively. Care must be taken when Grassmann differentiation of a trace of Grassmann operators is involved. The results for fermion operators make use of the anticommutation features of fermion operators and Grassmann variables and the rules for Grassmann left and right differentiation. It is important to move the Grassmann variable being differentiated to the left of the expression to implement left differentiation, and to the extreme right for carrying out right differentiation. These movements involve the use of anticommutation rules for pairs of Grassmann variables or a Grassmann variable and a fermion operator, plus of course the anticommutation rules in (2.18) for pairs of fermion operators. Grassmann variables and fermion operators commute with c-numbers and boson operators.
The derivation of the correspondence rules involves considering differentiating the product of two functions and , each of which is either even or odd. Results for differentiating products of functions that are neither even nor odd can be obtained by first expressing each function as the sum of even and odd components and then using the linearity property of differentiation. The product rules are
where or depending on whether e or f is even or odd.
The integration-by-parts result for Grassmann functions can be derived by applying the product differentiation rules and noting that the full phase space Grassmann integral of a Grassmann derivative is zero. We have
Some other rules needed are those for differentiating the Grassmann integral over all of a function of two sets of Grassmann variables and . The Grassmann integral will be a function of the . The rule is that differentiation from either direction after integration gives the same result as differentiating first and then integrating. Thus we have
(p.363) In addition, certain rules involving the traces of Grassmann operators are needed. These include the cyclic rules
which apply when the two Grassmann operators are either both even or both odd. The rules for differentiating or multiplying the traces of Grassmann operators are also needed. For differentiation, these are
with analogous results involving multiplication. The rules require two differentiation or multiplication processes to occur – no simple rules exist for just a single process.
F.1.2 Boson Case – Canonical-Density-Operator Approach
Let us introduce, for simplicity of notation, the fermionic operator
which clearly commutes with . Hence we can write the density operator in two alternative forms as
does not involve any bosonic operators.
Certain results involving the bosonic projector can be obtained from Chapter 5, which give and . We use the fact that depends only on the and , and that both and commute with any c-number. Moreover, depends only on the and depends only on the , and it follows that is an analytic function of the and . It therefore can be differentiated with respect to the and , and differentiation with respect to either or and either or yields the same result in the two cases. We find that
(p.364) where the standard product differentiation rules for complex functions have been used. By way of illustration, we obtain the third of these as follows:
We can now apply these results to determine the changes to the distribution function. For example, if is replaced by , then
where we have used integration by parts, assuming that the non-analytic function goes to zero on the boundary of the phase space. In addition, as and hence are not analytic functions, differentiation with respect to or gives different results and hence the interpretation of differentiation with respect to is taken to mean that one of these choices has been made. Thus, as is a valid distribution function, we see that
(p.365) The other three cases are proved similarly. To summarise the results,
F.1.3 Boson Case – Characteristic-Function Approach
The characteristic function may be written as
where we have incorporated the density operator and the Grassmann fermion operators into the single operator
We shall normally leave the dependence implicit. Note that is a Grassmann operator that involves the full Bose–Fermi density operator and fermion annihilation and creation operators.
Some bosonic identities will be useful:
These are used, in particular, to determine how the characteristic function changes when is replaced by etc. For example, if is replaced by , then becomes . Then, denoting that with , we have , we find
(p.366) where we have made use of the cyclic property of the trace.
The other cases may be derived in a similar fashion.
Summarising the results, we have the correspondence rules:
Our next task is to translate these into correspondence rules for the distribution function. We recall that the characteristic and distribution functions are related via
where we have introduced the simplifying notation
which is, of course, a Grassmann function of , although this will be left implicit. As an example of the procedure, if is replaced by then the characteristic function becomes
We note that is an analytic function of , and it follows that
Applying these to (F.37) and performing integration by parts, we recover the associated transformation:
(p.367) Note that the function is not an analytic function of , but it is still differentiable with respect to and . Hence only derivatives with respect to these variables may be used in performing the integration-by-parts step, which also relies on and hence going to zero fast enough on the boundaries of the planes to allow boundary terms to be ignored. It follows that the change to the distribution function is as follows:
The other cases
may be derived in a similar fashion to give the correspondence rules:
The symmetrically ordered case can be obtained from the normally ordered case, since the two characteristic functions are related as in (7.10):
We can use this relationship together with the results for the normally ordered case to determine the change in the symmetrically ordered characteristic function and hence in the corresponding distribution function. For example, if is replaced by , we see that
It follows, therefore, that pre-multiplying the density operator by corresponds to the transformation
(p.368) The other cases are obtained in a similar manner to give
It is straightforward to apply these to the relationship between the symmetrically ordered characteristic function and the double-space Wigner distribution function to derive the correspondence rules for the Wigner distribution function. We find
F.1.4 Fermion Case – Density Operator Approach
Let us begin, in order to condense the expressions that arise, by introducing the operator
We note that this is both a bosonic operator and also an even Grassmann function and that, because it is even, commutes with . It follows that we can write the density operator in two alternative forms,
Note that does not involve any fermionic operators.
Certain results involving the fermion projector can be obtained from Chapter 5, which give and . We use the fact that (p.369) is an even Grassmann function, and that both and will commute with any Grassmann number, as they just involve factors like or where two anticommuting quantities are involved. Also, depends only on the and depends only on the gi. These are neither even nor odd functions, however, although is an even Grassmann operator. We find that
where the product differentiation rules (4.28) have been used. By way of illustration, we obtain the third of these as follows:
All of the above results can also be written in another form, with the Grassmann number differentiation or multiplication applied from the other direction. The Grassmann Bargmann projector is an even Grassmann operator and behaves like an even Grassmann function, as if it is expanded out, there are always an even number of anticommuting quantities in every term. This is because either the Grassmann numbers appear in pairs, as in the factor, or they appear in a pair with a fermion annihilation or creation operator, as in the expressions for or . In regard to multiplication by a Grassmann number gi or , there are always an even number of anticommutations involved in taking the number from one side of . Also, as mentioned before, the proof of the result (4.21) that left and right differentiation of an even Grassmann function are related by a factor can be extended to the case of an even Grassmann operator because, in the movement of the Grassmann number to the left or right for differentiation, it would not matter if fermion annihilation or creation operators replaced some of the Grassmann numbers, (p.370) since they would still anticommute with the Grassmann number being differentiated. Hence we also have
where the sign of the derivative changes but not that of the Grassmann number.
We can apply these results to determine the changes to the distribution function. For example, if is replaced by , then
where we have used the third of the integration-by-parts rules (F.9), noting that is an even Grassmann operator.
As is a valid distribution function, we see that
where the last step is based on (4.21) and depends on being an even Grassmann function. The other cases may be treated in a similar manner. The overall results are
(p.371) The symmetry between the first forms for the results is to be noted, with a related symmetry applying to the second forms.
F.1.5 Fermion Case – Characteristic-Function Method
The characteristic function is related to the distribution function via Grassmann and c-number integrals:
One way of justifying the correspondence rules is to assume their effect on the distribution function and derive their effect on the characteristic function, given that the latter is required to be related to the distribution function via the phase space integral form (7.51). We can then return to the basic definition of the characteristic function (7.50) and confirm that the new characteristic function does correspond to the characteristic function that would apply when the density operator is replaced by its product with a fermion annihilation or creation operator. However, the process of multiplying or differentiating a trace with Grassmann variables is involved, and to carry out these processes inside the trace requires there to be two Grassmann processes involved (see (5.137)). Hence the correspondence rules can only be confirmed in this way for cases where the density operator is replaced by its product with two fermion annihilation and/or creation operators. Unfortunately, though, there are a total of 12 different replacements that might be considered, corresponding to pre-multiplication by two operators, post-multiplication by two operators or sandwiching the density operator between two operators. In terms of applying the correspondence rules in physical situations, we note that kinetic-energy and one-body interaction terms in the Hamiltonian always involve pairs of fermion operators , two-body interaction terms involve two pairs , terms in master equations involve forms such as , and so on. Even in the case of combined boson–fermion systems where pairs of fermions may be created or destroyed, the terms in the Hamiltonian will involve pairs of annihilation or creation operators. All these require the density operator to replaced in succession by a product of the density operator with two fermion annihilation or creation operators. Hence, in physical situations it is not necessary to consider the replacement of the density operator by its product with only one fermion annihilation or creation operator.
A second approach involving the characteristic function is to consider the effect of multiplying the density operator by fermion annihilation or creation operators within the definition (7.11) of the characteristic function and then use its relationship to the distribution function to determine the correspondence rule. However, the same problems as described in the last paragraph also occur, and this approach also requires consideration of the effect of pairs of fermion operators on the density operator – as we will now illustrate.
(p.372) As an example of the second approach to deriving the fermion correspondence rules for pairs of fermion operators from the characteristic function, consider the case when . Because pair cases are complicated, we will ignore the boson operators and variables. In this case
It is straightforward to show, using the anticommutation rules, that
(p.373) so that
This is the sum of four terms
Note that we are not yet in a position to take Grassmann variables outside the trace.
Using the cyclic properties of the trace and the fermion anticommutation rule, we have
It is then straightforward to show that
where we have used the result for double Grassmann differentiation of a trace.
Using a similar approach, we can show that
Collecting these terms together, we have
showing that the overall effect is to take the original characteristic function and apply to it an even number of Grassmann derivatives or multipliers.
We now express the characteristic function in terms of the distribution function,
(p.374) and carry out the required operations on it. We note that
On performing the Grassmann differentiations, we then find
So, after moving the Grassmann variables, we get
Next we use Grassmann integration by parts to move the derivatives onto the distribution function to give
Finally, we can simplify this by making use of the following:
(p.376) In this way, we arrive at
which we recognise as the correct correspondence rule for .
F.1.6 Boson Case – Canonical-Distribution Rules
To obtain the bosonic canonical correspondence rules from the standard correspondence rules, we note that
Using (8.8), we see that
F.2 Successive Correspondence RulesF.2
We wish to consider situations where is replaced by its product with several boson or fermion operators. To proceed further it is convenient to consider a general operator , where as appropriate. In general, involves cases where is replaced by successive products. As in the single-operator case, (p.377) we can determine the outcome via the use of canonical forms for the density operator or using the characteristic-function approach. We can define a general characteristic function via
which is related to a general distribution function via
The effect of the successive multiplications on the characteristic function can be determined via their effect on the factors and followed by taking the trace (fermion operators must be taken in pairs) and then working out the effect on the distribution function using (F.77).
The general operator will have a canonical representation in the standard form
are normalised projectors, and and are the usual fermionic and bosonic Bargmann projectors. The effect of successive multiplications with annihilation and creation operators can be determined by first applying the process to the Bargmann projectors using (8.29) and (8.33). These operators depend analytically on the and are even Grassmann functions of the . The use of integration by parts then enables the effect on the canonical distribution function to be determined.
For the more general result in which an existing product of the density operator with boson and fermion operators is replaced by times a single fermion annihilation or creation operator, the existing distribution function is replaced as follows:
(p.378) where if is an even Grassmann function and if is odd.
The safest way to proceed for fermion operators is always to use the result which does not depend on whether is even or odd:
For bosonic systems, a similar analysis gives
There are many options in applying the boson correspondence rules, owing to the analytic features of Bargmann projectors or characteristic functions.