## Bryan J. Dalton, John Jeffers, and Stephen M. Barnett

Print publication date: 2014

Print ISBN-13: 9780199562749

Published to Oxford Scholarship Online: April 2015

DOI: 10.1093/acprof:oso/9780199562749.001.0001

Show Summary Details
Page of

PRINTED FROM OXFORD SCHOLARSHIP ONLINE (www.oxfordscholarship.com). (c) Copyright Oxford University Press, 2017. All Rights Reserved. Under the terms of the licence agreement, an individual user may print out a PDF of a single chapter of a monograph in OSO for personal use (for details see http://www.oxfordscholarship.com/page/privacy-policy). Subscriber: null; date: 26 February 2017

# Appendix F Fokker–Planck Equations

Source:
Phase Space Methods for Degenerate Quantum Gases
Publisher:
Oxford University Press

# (p.360) F.1 Correspondence Rules

## F.1.1 Grassmann and Operator Formulae

In the Liouville–von Neumann equation, the density operator $ρˆ$ may be replaced by $Ξˆρˆ$ or $ρˆΞˆ$ or $ΞˆρˆΓˆ$, where $Ξˆ$ and $Γˆ$ 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 $χ(ξ,ξ+,h,h+)$ is

(F.1)
$Display mathematics$

where $ξ≡{ξ1,ξ2,⋯,ξp}$ and $ξ+≡{ξ1+,ξ2+,⋯,ξp+}$ are two sets of c-numbers associated with the p bosonic modes, and $h≡{h1,h2,⋯,hn}andh+≡{h1+,h2+,⋯,hn+}$ are two sets of Grassmann numbers associated with the n fermionic modes.

The characteristic function and the distribution function $P(α,α+,α∗,α+∗,g,g+)$ are related via Grassmann and c-number integrals:

(F.2)
$Display mathematics$

where $α≡{α1,α2,⋯,αp}andα+≡{α1+$ and $α2+,⋯,αp+}$ are also two sets of c-numbers associated with the p bosonic modes, and $g≡{g1,g2,⋯,gn}$ and $g+≡{g1+,g2+,⋯,gn+}$ 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:

(F.3)
$Display mathematics$

where

(F.4)
$Display mathematics$

is a product of normalised projectors, with $Λˆf(g,g+)$ and $Λˆb(α,α+)$ being the usual fermion and boson Bargmann state projectors:

(F.5)
$Display mathematics$

The two forms for the density operator apply because $Λˆ(g,g+,α,α+)$ is an even Grassmann operator and therefore commutes with $Pcanon(α,α+,α∗,α+∗,g,g+)$.

(p.362) The derivation of the correspondence rules involves using standard quantum rules such as

(F.6)
$Display mathematics$

involving operators $Ξˆ$ and $Sˆ$ 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 $e(g)$ and $f(g)$, 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

(F.7)
$Display mathematics$
(F.8)
$Display mathematics$

where $σ(e),σ(f)=+1$ or $−1$ 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

(F.9)
$Display mathematics$
(F.10)
$Display mathematics$
(F.11)
$Display mathematics$
(F.12)
$Display mathematics$

Some other rules needed are those for differentiating the Grassmann integral over all $gi,gi+$ of a function $F(g,g+,h,h+)$ of two sets of Grassmann variables ${gi,gi+}$ and ${hj,hj+}$. The Grassmann integral will be a function of the $hj,hj+$ . The rule is that differentiation from either direction after integration gives the same result as differentiating first and then integrating. Thus we have

(F.13)
$Display mathematics$
(F.14)
$Display mathematics$
(F.15)
$Display mathematics$
(F.16)
$Display mathematics$

(p.363) In addition, certain rules involving the traces of Grassmann operators are needed. These include the cyclic rules

(F.17)
$Display mathematics$
(F.18)
$Display mathematics$

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

(F.19)
$Display mathematics$

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

(F.20)
$Display mathematics$

which clearly commutes with $Λˆb(α,α+)$. Hence we can write the density operator in two alternative forms as

(F.21)
$Display mathematics$

Note that $Πˆf$

does not involve any bosonic operators.

Certain results involving the bosonic projector can be obtained from Chapter 5, which give $aˆi|α〉B=αi|α〉B,B〈α+∗|aˆi†=B〈α+∗|αi+,aˆi†〈α|B=(∂/∂αi)〈α|B$ and $B〈α+∗|aˆi=(∂/∂αi+)B〈α+∗|$. We use the fact that $Tr(|α〉BB〈α+∗|=exp(α⋅α+)$ depends only on the $αi$ and $αi+$, and that both $|α〉B$ and $B〈α+∗|$ commute with any c-number. Moreover, $B〈α+∗|$ depends only on the $αi+$ and $〈α|B$ depends only on the $αi$, and it follows that $Λˆb(α,α+)$ is an analytic function of the $αi$ and $αi+$. It therefore can be differentiated with respect to the $αi$ and $αi+$, and differentiation with respect to either $αix$ or $iαiy$ and either $αix+$ or $iαiy+$ yields the same result in the two cases. We find that

(F.22)
$Display mathematics$

(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:

(F.23)
$Display mathematics$

We can now apply these results to determine the changes to the distribution function. For example, if $ρˆ$ is replaced by $aˆi†ρˆ$, then

(F.24)
$Display mathematics$

where we have used integration by parts, assuming that the non-analytic function $P(α,α+,α∗,α+∗,g,g+)$ goes to zero on the boundary of the $α,α+$ phase space. In addition, as $P(α,α+,α∗,α+∗,;g,g+)$ and hence $Πˆf$ are not analytic functions, differentiation with respect to $αix$ or $iαiy$ gives different results and hence the interpretation of differentiation with respect to $αi$ is taken to mean that one of these choices has been made. Thus, as $P(α,α+,α∗,α+∗,g,g+)$ is a valid distribution function, we see that

(F.25)
$Display mathematics$

where

(F.26)
$Display mathematics$

(p.365) The other three cases are proved similarly. To summarise the results,

(F.27)
$Display mathematics$

where

(F.28)
$Display mathematics$

## F.1.3 Boson Case – Characteristic-Function Approach

The characteristic function may be written as

(F.29)
$Display mathematics$

where we have incorporated the density operator and the Grassmann fermion operators into the single operator

(F.30)
$Display mathematics$

We shall normally leave the $h,,h+$ dependence implicit. Note that $Θˆ(h,\,h+)$ is a Grassmann operator that involves the full Bose–Fermi density operator and fermion annihilation and creation operators.

Some bosonic identities will be useful:

(F.31)
$Display mathematics$
(F.32)
$Display mathematics$

These are used, in particular, to determine how the characteristic function changes when $ρˆ$ is replaced by $aˆiρˆ$ etc. For example, if $ρˆ$ is replaced by $aˆi†ρˆ$, then $Ωˆb+(ξ+)Θˆ(h,\,h+)×Ωˆb−(ξ)$ becomes $Ωˆb+(ξ+)aˆi†Θˆ(h,\,h+)Ωˆb−(ξ)$. Then, denoting that with $Sˆ=iaˆ⋅ξ+$, we have $[Sˆ,aˆi†]=iξi+,[Sˆ,[Sˆ,aˆi†]]=0,⋯$, we find

(F.33)
$Display mathematics$

(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:

(F.34)
$Display mathematics$

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

(F.35)
$Display mathematics$

where we have introduced the simplifying notation

(F.36)
$Display mathematics$

which is, of course, a Grassmann function $Θ(α,α+,ξ∗,ξ+∗,\,h,\,h+)$ of $h,\,h+$, although this will be left implicit. As an example of the procedure, if $ρˆ$ is replaced by $aˆi†ρˆ$ then the characteristic function becomes

(F.37)
$Display mathematics$

We note that $exp(iα⋅ξ+)$ is an analytic function of $αi$, and it follows that

(F.38)
$Display mathematics$

Applying these to (F.37) and performing integration by parts, we recover the associated transformation:

(F.39)
$Display mathematics$

(p.367) Note that the function $Θ(α,α+,α∗,α+∗)$ is not an analytic function of $αi$, but it is still differentiable with respect to $αix$ and $αiy$. Hence only derivatives with respect to these variables may be used in performing the integration-by-parts step, which also relies on $P(α,α+,α∗,α+∗,g,\,g+)$ and hence $Θ(α,α+,α∗,α+∗)$ going to zero fast enough on the boundaries of the $αi$ planes to allow boundary terms to be ignored. It follows that the change to the distribution function is as follows:

(F.40)
$Display mathematics$

The other cases

may be derived in a similar fashion to give the correspondence rules:

(F.41)
$Display mathematics$

The symmetrically ordered case can be obtained from the normally ordered case, since the two characteristic functions are related as in (7.10):

(F.42)
$Display mathematics$

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 $ρˆ$ is replaced by $aˆiρˆ$, we see that

(F.43)
$Display mathematics$

It follows, therefore, that pre-multiplying the density operator by $aˆi$ corresponds to the transformation

(F.44)
$Display mathematics$

(p.368) The other cases are obtained in a similar manner to give

(F.45)
$Display mathematics$

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.46)
$Display mathematics$

## F.1.4 Fermion Case – Density Operator Approach

Let us begin, in order to condense the expressions that arise, by introducing the operator

(F.47)
$Display mathematics$

We note that this is both a bosonic operator and also an even Grassmann function and that, because it is even, $Πˆb$ commutes with $Λˆf(g,\,g+)$. It follows that we can write the density operator in two alternative forms,

(F.48)
$Display mathematics$

Note that $Πˆb(g,g+)$ does not involve any fermionic operators.

Certain results involving the fermion projector can be obtained from Chapter 5, which give $cˆi|g〉B=gi|g〉B,B〈g+∗|cˆi†=B〈g+∗| gi+,cˆi† |g|B=(−∂→/∂gi)|g〉B$ and $B〈g+∗|cˆi=B〈g+∗|(−∂←/∂gi+)$. We use the fact that $Tr(|g〉BB〈g+∗|)=exp(g⋅g+)$ (p.369) is an even Grassmann function, and that both $|g〉B$ and $B〈g+∗|$ will commute with any Grassmann number, as they just involve factors like $cˆi†gi$ or $cˆigi+$ where two anticommuting quantities are involved. Also, $B〈g+∗|$ depends only on the $gi+$ and $|g⟩B$ depends only on the gi. These are neither even nor odd functions, however, although $Λˆf(g,\,g+)$ is an even Grassmann operator. We find that

(F.49)
$Display mathematics$

where the product differentiation rules (4.28) have been used. By way of illustration, we obtain the third of these as follows:

(F.50)
$Display mathematics$

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 $Λˆf(g,\,g+)$ 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 $exp(−g⋅g+)$ factor, or they appear in a pair with a fermion annihilation or creation operator, as in the expressions for $|g〉B$ or $〈g+∗|B$. In regard to multiplication by a Grassmann number gi or $gi+$, there are always an even number of anticommutations involved in taking the number from one side of $Λˆf(g,\,g+)$. 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 $−1$ 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

(F.51)
$Display mathematics$

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 $ρˆ$ is replaced by $cˆi†ρˆ$, then

(F.52)
$Display mathematics$

where we have used the third of the integration-by-parts rules (F.9), noting that $Λˆf(g,g+)$ is an even Grassmann operator.

As is a valid distribution function, we see that

(F.53)
$Display mathematics$

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

(F.54)
$Display mathematics$

(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:

(F.55)
$Display mathematics$

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 $ρˆ$ 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 $ρˆ$ 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 $cˆi†cˆj$, two-body interaction terms involve two pairs $cˆi†cˆj†cˆkcˆl$, terms in master equations involve forms such as $[cˆi†cˆj,ρˆ],[cˆi†cˆj†cˆkcˆl,ρˆ],[cˆi†,ρˆcˆj]$, $[cˆi,ρˆcˆj†]$ 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 $ρˆ$ 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 $ρˆ⇒cˆi†ρˆcˆj$. Because pair cases are complicated, we will ignore the boson operators and variables. In this case

(F.56)
$Display mathematics$

It is straightforward to show, using the anticommutation rules, that

(F.57)
$Display mathematics$

(p.373) so that

(F.58)
$Display mathematics$

This is the sum of four terms

(F.59)
$Display mathematics$

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

(F.60)
$Display mathematics$

It is then straightforward to show that

(F.61)
$Display mathematics$

so that

(F.62)
$Display mathematics$

where we have used the result for double Grassmann differentiation of a trace.

Using a similar approach, we can show that

(F.63)
$Display mathematics$

Collecting these terms together, we have

(F.64)
$Display mathematics$

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,

(F.65)
$Display mathematics$

(p.374) and carry out the required operations on it. We note that

(F.66)
$Display mathematics$

On performing the Grassmann differentiations, we then find

(F.67)
$Display mathematics$

So, after moving the Grassmann variables, we get

(F.68)
$Display mathematics$

(p.375) and, using the results in (F.66), we then have

(F.69)
$Display mathematics$

Next we use Grassmann integration by parts to move the derivatives onto the distribution function to give

(F.70)
$Display mathematics$

Finally, we can simplify this by making use of the following:

(F.71)
$Display mathematics$

(p.376) In this way, we arrive at

(F.72)
$Display mathematics$

which we recognise as the correct correspondence rule for $ρˆ→cˆi†ρˆcˆj$.

## F.1.6 Boson Case – Canonical-Distribution Rules

To obtain the bosonic canonical correspondence rules from the standard correspondence rules, we note that

(F.73)
$Display mathematics$

and that

(F.74)
$Display mathematics$

Using (8.8), we see that

(F.75)
$Display mathematics$

When these are applied to the canonical form (8.15) for the distribution function, the correspondence rules (8.18) follow. The effect of the operators involving $δi$, $δi∗$ gives zero.

# F.2 Successive Correspondence RulesF.2

We wish to consider situations where $ρˆ$ is replaced by its product with several boson or fermion operators. To proceed further it is convenient to consider a general operator $σˆ$, where $σˆ=ρˆ,cˆiρˆ,ρˆcˆi,cˆi†ρˆ,ρˆcˆi†,cˆiρˆcˆj†,cˆj†cˆiρˆ,ρˆcˆj†cˆi,⋯$ as appropriate. In general, $σˆ$ involves cases where $ρˆ$ 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 $κ(ξ,ξ+,h,h+)$ via

(F.76)
$Display mathematics$

which is related to a general distribution function $S(α,α+,α∗,α+∗,g,g+)$ via

(F.77)
$Display mathematics$

The effect of the successive multiplications on the characteristic function can be determined via their effect on the factors $Ωˆb+(ξ+)Ωˆf+(h+)$ and $Ωˆf−(h)Ωˆb−(ξ)$ 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 $σˆ$ will have a canonical representation in the standard form

(F.78)
$Display mathematics$

where

(F.79)
$Display mathematics$

are normalised projectors, and $Λˆf(g,g+)$ and $Λˆb(α,α+)$ 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 $α,α+$ and are even Grassmann functions of the $g,g+$. 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 $σˆ$ of the density operator $ρˆ$ with boson and fermion operators is replaced by $σˆ$ times a single fermion annihilation or creation operator, the existing distribution function $S(α,α+,α∗,α+∗,g,g+)$ is replaced as follows:

(F.80)
$Display mathematics$
(F.81)
$Display mathematics$
(F.82)
$Display mathematics$
(F.83)
$Display mathematics$
(F.84)
$Display mathematics$
(F.85)
$Display mathematics$
(F.86)
$Display mathematics$
(F.87)
$Display mathematics$

(p.378) where $e(S)=+1$ if $S(α,α+,α∗,α+∗,g,g+)$ is an even Grassmann function and $e(S)=−1$ if $S(α,α+,α∗,α+∗,g,g+)$ is odd.

The safest way to proceed for fermion operators is always to use the result which does not depend on whether $S(α,α+,α∗,α+∗,g,g+)$ is even or odd:

(F.88)
$Display mathematics$
(F.89)
$Display mathematics$
(F.90)
$Display mathematics$
(F.91)
$Display mathematics$

For bosonic systems, a similar analysis gives

(F.92)
$Display mathematics$
(F.93)
$Display mathematics$
(F.94)
$Display mathematics$
(F.95)
$Display mathematics$

There are many options in applying the boson correspondence rules, owing to the analytic features of Bargmann projectors or characteristic functions.