Some reference formulas for the generating functions of canonical transformations
Abstract
We study some properties of the canonical transformations in classical mechanics and quantum field theory and give a number of practical formulas concerning their generating functions. First, we give a diagrammatic formula for the perturbative expansion of the composition law around the identity map. Then we propose a standard way to express the generating function of a canonical transformation by means of a certain “componential” map, which obeys the Baker–Campbell–Hausdorff formula. We derive the diagrammatic interpretation of the componential map, work out its relation with the solution of the Hamilton–Jacobi equation and derive its timeordered version. Finally, we generalize the results to the Batalin–Vilkovisky formalism, where the conjugate variables may have both bosonic and fermionic statistics, and describe applications to quantum field theory.
Keywords
Canonical Transformation Jacobi Equation Perturbative Expansion Gauge Fermion Background Field Method1 Introduction
Canonical transformations have a variety of applications, from classical mechanics to quantum field theory. In particular, they play an important role when quantum field theory is formulated by means of the functional integral and the Batalin–Vilkovisky (BV) formalism [1, 2, 3]. The BV formalism associates external sources \(K_{\alpha }\) with the fields \(\Phi ^{\alpha }\) and introduces a notion of antiparentheses (X, Y) of functionals X, Y of \(\Phi \) and K. This formal setup is convenient to treat general gauge theories and study their renormalization, because it collects the Ward–Takahashi–Slavnov–Taylor (WTST) identities [4, 5, 6, 7] in a compact form and relates in a simple way the identities satisfied by the classical action \(S(\Phi ,K)\) to the identities satisfied by the generating functional \(\Gamma \) of the oneparticle irreducible correlation functions. The canonical transformations, which are the field/source redefinitions that preserve the antiparentheses, appear in several contexts. For example, they provide simple ways to gaugefix the theory and map the WTST identities under arbitrary changes of field variables and gaugefixing. Moreover, they are a key ingredient of the subtraction of divergences.
The generating functionals of the canonical transformations used in quantum field theory are often polynomial, and can be composed and inverted with a small effort. Nevertheless, there are exceptions. When the theory is nonrenormalizable, for example, as the standard model coupled to quantum gravity, the canonical transformations involved in the subtraction of the divergences are nonpolynomial and arbitrarily complicated. Even when the theory is power counting renormalizable, the variety of fields and sources that are present and their statistics make it useful to have some shortcuts and practical formulas to handle the basic operations on canonical transformations in more straightforward ways.
In this paper, we collect a number of reference formulas concerning the generating functions of canonical transformations and give diagrammatic interpretations of their perturbative versions. We first work in classical mechanics and then generalize the investigation to the BV formalism. The generalization is actually straightforward, since the operations we define preserve the statistics of the functionals.
In Sect. 2 we start from the composition law, by writing the generating function of the composed canonical transformation as the treelevel projection of a suitable functional integral. So doing, the perturbative expansion of the result around the identity map can easily be expressed in a diagrammatic form. In Sect. 3 we relate the composition law to the Baker–Campbell–Hausdorff (BCH) formula [8, 9, 10, 11, 12, 13]. We propose a standard way of expressing the generating function of a canonical transformation by means of a componential map \(\mathcal {C}(X)\) such that \(\mathcal {C}^{1}(X)=\mathcal {C}(X)\) and \(\mathcal {C}^{1}(\mathcal {C} (X)\circ \mathcal {C}(Y))=\) BCH(X, Y). In Sect. 4 we derive the relation between the componential map and the solution of the Hamilton–Jacobi equation for timeindependent Hamiltonians. In Sect. 5 we work out the diagrammatic interpretation of the perturbative expansion of the componential map around the identity map. In Sect. 6 we generalize the formulas to timedependent Hamiltonians, which gives the timeordered version of the componential map. In Sect. 7 we extend the analysis to the BV formalism, where the fields can have arbitrary statistics. We illustrate a number of applications to quantum field theory. Section 8 contains the conclusions.
2 Composition of canonical transformations
In this section we study the composition of canonical transformations. We first recall the basic formulas for the generating function of the composite canonical transformation, in terms of the generating functions of the components. Then we express the result as the treelevel sector of a functional integral and provide a diagrammatic interpretation of its perturbative expansion around the identity map.
In this paper, we are mostly interested in formulas that may have practical uses in perturbative quantum field theory. It is more convenient to describe the canonical transformations \(q,p\rightarrow Q,P\) by means of generating functions of the form F(q, P), rather than G(q, Q), because the former can easily be expanded around the identity transformation and allow us to express the composite canonical transformation diagrammatically. It is not possible to achieve these goals in a simple way with generating functions of the form G(q, Q).
By definition, we include the diagrams that have no lines, that is to say, the vertex u and the vertex v. The number of vertices is called order of the diagram. The absence of loops implies that a diagram of order n contains \(n1\) lines, with \(n\geqslant 1\). Note that there are no external legs.
3 The componential map
The composition law of the previous section is good for a number of purposes, but not practical in other cases. For example, it does not provide a simple way to invert a canonical transformation. In this section, we propose a standard way of expressing the generating function of a canonical transformation by means of a “componential” map and rephrase the composition law in a way that makes various properties more apparent. The componential map is written as a perturbative expansion around the identity map and obeys the BCH formula. Among other things, it makes the inverse operation straightforward.
4 Relation with the solution of the Hamilton–Jacobi equation
As promised, the componential map is uniquely determined by the composition law. However, we still have to prove that formula (3.2) holds for arbitrary X and Y. This goal can be achieved by working out the relation between the componential map and the solution of the Hamilton–Jacobi equation.
Setting \(\mathcal {C}(Y)=F_{A}\), \(\mathcal {C}(X)=F_{B}\), and \(F_{C}=\mathcal {C} (X)\circ \mathcal {C}(Y)\), we can easily check the first few orders of (3.2) by comparing the formulas (2.15) and (3.7).
Summarizing, the componential map is a sort of generating function for the exponential map. Indeed, the transformations of the coordinates and the momenta are given by the exponential map and generated by the componential map.
5 Diagrammatics of the componential map
 (i)
divide by the number of permutations of the identical subdiagrams \( G_{mi}^{\prime }\), \(m<n\);
 (ii)
multiply by the number of ways to obtain each subdiagram \( G_{mi}^{\prime }\), \(m<n\), by attaching the extra incoming line to \(G_{mi}\);
 (iii)
multiply by the coefficients \(e_{mi}\) of the subdiagrams \(G_{mi}\), \( m<n \).
Formula (5.7) and the rules just listed are straightforward consequences of (5.3). We have decomposed the diagram \(G_{nj}\) into its contributions as they appear on the righthand side of (5.3), which are the cut diagrams \(G_{njk}^{\text {cut}}\). Each of them has a simple combinatorial factor \(e_{njk}\). The sum of those combinatorial factors, divided by n, gives \(e_{nj}\).
An alternative, actually simpler, way to work out the diagrammatic expansion of the componential map is given in the next section. It follows from the expansion of the timeordered componential map, which has straightforward coefficients. The coefficients of \(\mathcal {C}(X)\) are the values of simple integrals that appear when the timeordered formula is specialized to the case of a timeindependent function X.
6 Timeordered componential map
A canonical transformation continuously connected to the identity can be viewed as a fictitious “time” evolution associated with a suitable “Hamiltonian”. This allows us to relate the componential map to the solution of the Hamilton–Jacobi equation. In Sect. 4 we have taken advantage of this correspondence in the case of timeindependent Hamiltonians, or, equivalently, \(\eta \)independent functions X(q, P). Generalizing the formulas of Sect. 4 to timedependent Hamiltonians H(q, p, t), we can obtain the timeordered (precisely, \(\eta \)ordered) componential map.

the disk with coordinate \(\eta _{k}\) is anterior (posterior) to the disk with coordinate \(\eta _{k^{\prime }}\) if \(\eta _{k}<\eta _{k^{\prime }}\) (\( \eta _{k}>\eta _{k^{\prime }}\));

a pair of disks is \(\eta \)ordered if one of them is anterior to the other;

two disks \(D_{1}\) and \(D_{2}\) are separated if the path connecting them (drawn by covering each line only once) contains a third disk \(D_{3}\) that is posterior to both;

the latest disk is the one with coordinate \(\eta _{k}\) such that \(\eta _{k}>\eta _{k^{\prime }}\) for every \(k^{\prime }\ne k\);

given a disk D, the disk \(D^{\prime }\) following D is the most anterior disk among the disks that are posterior to D and not separated from D.
Then, construct all the inequivalent diagrams. Call them \(\tilde{G}_{nj}\), where n is the number of disks and j is an extra label. Denote the set of diagrams with n disks by \(\mathcal {\tilde{D}}_{n}\).
7 Canonical transformations and Batalin–Vilkovisky formalism
In this section we generalize the results found so far to the Batalin–Vilkovisky formalism, where the generating function(al)s are fermionic and the fields may be both bosonic and fermionic. Then we give some examples that have applications to both renormalizable and nonrenormalizable theories. We compose the canonical transformations that perform the gaugefixing with those that switch to the background field method. Then we use the componential map to interpolate between the background field approach and the standard nonbackground approach.
The fields \(\Phi ^{\alpha }\) include the classical fields \(\phi ^{i}\), the Fadeev–Popov ghosts \(C^{I}\), the antighosts \(\bar{C}^{I}\) and the Lagrange multipliers \(B^{I}\) for the gaugefixing. The action \(S(\Phi ,K)\) is a local functional that satisfies the master equation \((S,S)=0\) and coincides with the classical action \(S_{c}(\phi )\) at \(C=\bar{C}=B=K=0\).
Canonical transformations are used for various purposes in quantum field theory. They encode the most general (changes of) gaugefixing and changes of field variables. Moreover, they are an important ingredient of the perturbative subtraction of divergences. Precisely, they subtract the divergences that are proportional to the field equations. The composition and the inversion of canonical transformations are operations that are met frequently. Often, it is enough to study them at the infinitesimal level, but sometimes it is necessary to handle them exactly or to all orders of the expansion. The literature on these topics is wide, both at the mathematical/formal level [1, 2, 18, 19] and at the level of renormalization and gauge dependence [20, 21, 22, 23, 24, 25, 26, 27, 28, 29].
We recall that the BV formalism is quite versatile and can be used to formulate all kinds of general gauge theories, including those where the symmetry transformations close only on shell and those that have reducible gauge algebras (where the ghosts have local gauge symmetries of their own and it is necessary to introduce “ghosts of ghosts”). Our formulas hold in those cases also.
After the shift, the action is \(F_{\text {b}}S\). The new fields \(\Phi ^{\alpha }\) are called quantum fields. The symmetry transformations \( R^{i}(\Phi )\) of \(\phi ^{i}\) are turned into the transformations \(R^{i}(\Phi +\underline{ \Phi })\) of \(\phi ^{i}+ \underline{ \phi } ^{i}\). These can be decomposed as the sum of the background transformations \(R^{i}( \underline{ \Phi })\) of \(\underline{ \phi } ^{i}\) plus the transformations \( R^{i}(\Phi +\underline{ \Phi })R^{i}( \underline{ \Phi })\) of \(\phi ^{i}\). In turn, the transformations of \(\phi ^{i}\) split into the sum of the quantum transformations of \(\phi ^{i}\) [made of the \(\underline{ C} \)independent part of \(R^{i}(\Phi + \underline{ \Phi } )R^{i}(\underline{ \Phi })\)], plus the background transformations of \(\phi ^{i}\) (the \(\underline{ C} \)dependent part). Something similar happens to the symmetry transformations of the ghosts C.
In Ref. [36] the tensor operator \(\zeta _{IJ}\) was set to zero, to make \(F_{\text {gf}}\) and \(F_{\text {nm}}\) commute. However, in some applications, such as the chiral dimensional regularization of Ref. [37], which is useful to treat nonrenormalizable general chiral gauge theories, it is necessary to keep \(\zeta _{IJ}\) nonvanishing, to have wellbehaved regularized propagators.
The gaugefixing is the last step of the construction of the action. Indeed, only after properly organizing the background transformations, it makes sense to talk about a background invariant gauge fermion. Thus, we must take \(F_{\text {gf}}\circ F_{\text {nm}}\), rather than \(F_{\text {nm}}\circ F_{\text { gf}}\).
The dependence of the correlation functions on the parameters introduced by a canonical transformation is encoded into the equations of gauge dependence [20, 21, 22, 25, 26, 27, 28, 38, 39, 40, 41], sometimes known as Nielsen identities. The componential map and the other tools of this paper may be convenient to manipulate those equations more efficiently. In particular, the interpolation (7.9) allows us to take advantage of the background field method and prove key properties of renormalization in simpler, more powerful ways. An illustration of this fact can be found in Ref. [42], where an important theorem about the cohomology of renormalization was proved. That theorem allows us to classify the structures of the counterterms and the local contributions to anomalies.
In turn, the classification of counterterms and anomalies is important to show, to all orders of the perturbative expansion, that the gauge symmetries are not affected by the subtraction of divergences (up to canonical transformations). The background field method and the interpolation (7.11) have been used [36] to achieve this goal in manifestly nonanomalous theories, renormalizable or not. In potentially anomalous nonrenormalizable theories, such as the standard model coupled to quantum gravity, which require a more involved regularization [37], the goal must be achieved together with the proof of the Adler–Bardeen theorem [43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54] for the cancelation of anomalies to all orders (when they vanish at one loop). Within the standard, nonbackground approach, this was done for the first time in Ref. [54]. The techniques of this paper and the results of [42] may be useful to upgrade the derivation of [54] to the background field approach and prepare the ground to make further progress.
8 Conclusions
Canonical transformations play an important role not only in classical mechanics, but also in quantum field theory. In several situations, it is useful to have practical formulas for the perturbative expansion of the generating functions around the identity map. In this paper we have given a number of such formulas, starting from the composition law, which we have expressed as the tree sector of a functional integral and later rephrased by means of the componential map.
The componential map is a standard way to express the generating function of a canonical transformation. It makes the inverse operation straightforward and obeys the Baker–Campbell–Hausdorff formula. It also admits a simple diagrammatic interpretation and a timeordered generalization. It can be related to the solution of the Hamilton–Jacobi equation, expressed as a perturbative expansion in powers of a suitable Hamiltonian, its derivatives and its integrals over time.
The formulas we have found can be straightforwardly generalized from classical mechanics to quantum field theory, where the functionals and the conjugate variables may have both bosonic and fermionic statistics. Particularly interesting are the applications to the Batalin–Vilkovisky formalism. Canonical transformations are commonly used to implement the gaugefixing, make arbitrary changes of field variables and changes of the gaugefixing itself, switch to the background field method and subtract the counterterms proportional to the field equations. Various times these operations must be composed and inverted. Practical formulas, such as the ones given in this paper, allow us to handle these operations quickly. In particular, they can be convenient in nonrenormalizable theories, where the cohomology of counterterms and anomalies involves nonpolynomial functionals and the renormalization of divergences involves nonpolynomial canonical transformations.
Footnotes
 1.
To our knowledge, very few textbooks report this property. One is Ref. [14], where it is ascribed to Hamilton. For a standard derivation, see also [15]. For a derivation from the semiclassical limit of quantum mechanics, see [16]. For elaborations from the point of view of symplectic groupoids, see [17].
 2.
 3.
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