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Book reviewed by Manfred Salmhofer
This book is based on the Aisenstadt lectures of Joel Feldman at the CRM in Montréal. It provides an exposition of techniques for the construction of interacting fermionic quantum field theories. The mathematical construction of a Fermi liquid by Feldman, Knörrer and Trubowitz has since then appeared in a series of 11 papers [1]. The book reviewed here can be considered as an introduction to these papers.
Functional integrals are notorious for being mathematically undefined and much effort has been made to change this situation. In quantum mechanics, the transition to imaginary time, i.e. replacing the time evolution operator by the heat kernel, leads to the Feynman-Kac formula which involves mathematically well-defined integrals with respect to Wiener measure (see, e.g. [2]). Similarly, the Wick rotation to imaginary time in quantum field theory gives a connection to statistical mechanics that generalizes the Feynman-Kac formula and allows for mathematically rigorous constructions of such theories. A Wightman theory in Minkowski space can be reconstructed provided the Osterwalder-Schrader axioms are satisfied by the Schwinger functions of the Euclidian theory [3]. Proving that the Schwinger functions associated to certain Lagrangians exist and satisfy these axioms is the program of constructive quantum field theory. Several books describe the progress in that direction [4,5]. These constructions are in general rather complicated because almost all models involve bosonic fields which require a combination of cluster expansions and large-field bounds in addition to renormalization.
In nonrelativistic theories corresponding to the quantum statistical mechanics of many-body systems, functional integrals have become a natural and very useful tool which by now is completely standard in theoretical solid-state physics. Theories of electrons with a short-range two-body interaction can be formulated as purely fermionic theories, which allow for a much simpler treatment that avoids cluster expansions. The book describes such techniques.
Physically interesting functions in second-quantized fermion systems
can be represented as fermionic functional integrals
over suitable Grassmann algebras.
For instance, the free energy density of a system of fermions
with Hamiltonian in a finite volume
at inverse temperature
The methods of constructive field theory, in particular the mathematical renormalization group (RG) method are well suited to treating this problem. Moreover, they give direct access to physical properties, namely equilibrium properties of such systems, without invoking Osterwalder-Schrader reconstruction theorems.
The constructive field theory approach to many-body systems was pioneered by Feldman and Trubowitz [6] and independently by Benfatto and Gallavotti [7], and has since then been used by these authors and others to prove a number of mathematical theorems about solid state systems which were not accessible to other methods. Variations of the same RG technique are being increasingly used as calculational tools in solid state theory [8,9].
Section 1.7 contains some facts about Wick ordering, an important technical ingredient. Wick ordering with respect to a Gaussian measure is simply introducing orthogonal polynomials for this measure, and the fermionic version is a straightforward variant of it. Wick ordering is not fundamental, but it simplifies some calculations and bounds, in particular the identification of ``overlapping loops'' [10], which are not discussed in this monograph but are crucial for the construction of the Fermi liquid in [1]. Wick ordering is also an important ingredient in the ring expansion developed in Part 2 of the book.
The treatment in all of Part 1 is for finite-dimensional algebras, while the object appearing on the right hand side of (2) is a formal, infinite-dimensional Grassmann integral. The calculus on finite-dimensional Grassmann algebras developed in part 1 extends in a natural way to infinite-dimensional ones if the covariance of the Gaussian integral is sufficiently regular. Ways to do this are discussed briefly in Appendix A. This extension does not play a central role in the construction because the integral with a singular covariance on the RHS of (2) has to be approached by taking a limit anyway, and because in most applications, one can construct it as a limit of finite-dimensional Grassmann integrals, to which the methods of Part 1 apply directly. For instance, for a lattice system, where is a finite set of cardinality , the dimension of the Grassmann algebra diverges as . Thus one needs to prove bounds that are uniform in .
For finite , the covariance is just an antisymmetric
matrix.
Section 1.8 contains important bounds for the analytical treatment
of the Grassmann integrals, the classic Gram estimate:
if an matrix has entries
where for all and ,
and are vectors in some Hilbert space , then
(4) |
The Gram estimate alone does, however, not imply
analyticity of the renormalization group map in
uniformly in .
For this, a sufficiently fast decay of the covariance is also
needed, i.e.
To prove uniform analyticity, one also needs to
exhibit certain connectedness properties of the logarithm
of ratios like the one in (3).
The connectedness property is stated most easily in terms
of a Feynman graph expansion: only connected graphs contribute.
It is this property that ensures that, e.g. the free energy density
defined in (1) indeed has a limit as
.
The full expansion into Feynman graphs cannot be used because
graph-by-graph estimates lead to zero radius of analyticity.
Connectedness can be made explicit in a way that allows
for convergent bounds in several ways.
In their work, the authors choose the ``ring expansion''.
In Section 2.2, they rewrite the Schwinger
functional
(Definition 2.1),
from which one can obtain the RG map ,
in terms of an operator acting on the Grassmann algebra as
The crucial hypotheses for the proof of analyticity, as formulated in Theorem 2.6 and Corollary 2.7, are in Definition 2.5. Hypothesis (HG) is a restatement of the Gram bound, and (HS) is the summability of the covariance given by (5).
In Section 2.4, some applications are given. The first example is the two-dimensional Gross-Neveu model. The covariance is the inverse of a Dirac operator. The interaction is a local four-fermion interaction. The limit to be taken is an ultraviolet limit, i.e. the singularity is in the short-distance behaviour of the covariance. This model is interesting because it is perturbatively just renormalizable, with power counting very similar to scalar theory in 4 dimensions, but a construction of this model has been possible [11,12] because it is UV asymptotically free.
After this, the many-fermion system in two spatial dimensions is treated. The big difference to the previous case is that the singularity of the many-fermion covariance in Fourier space is not pointlike but on a submanifold of codimension one, the Fermi surface (in two dimensions a curve). A treatment along the lines sufficient for point singularities is presented as a warmup. It gives bounds that are too weak to control the -behaviour in a good way. After that, the ``sectorization'', an angular decomposition of the Fermi curve, is introduced and a basic convergence theorem (Theorem 2.15 MB) is proven. The introduction of sectors allows to use geometrical restrictions posed by the shape of the Fermi curve, to improve the bounds such that physically interesting models can be constructed. The sectorization, first introduced in [13], is a central technical tool in the study of these systems in two dimensions. The bounds for the Gram constant and the decay constant of the covariance are given in Appendix C.
In all these applications, the bounds given in the book are restricted to those interaction terms that decrease under the iteration of the RG map (the irrelevant terms in RG language). That is, a straightforward iteration in of these bounds is not possible because of the hypothesis that the part of degree in the Grasssmann generators , , vanishes for small values of (e.g. for in Theorem 2.15 MB). Even if this holds for , it does not hold for , so one cannot iterate. In fact, the with small are those that tend to grow under iteration of the RG map, and they have to be controlled by a careful choice of counterterms and initial conditions, as well as a much more detailed analysis of the RG map and its iteration. This is a hard task which takes up most of [1] and other constructions of these models, and not treated in the book. The bounds in the book do imply that one can focus on the flow of the for small . These functions are also the ones that one restricts to in practical applications to physics.
Unfortunately, there is almost no discussion of how the results derived in the book fit into the general strategy of [1]. There is also no mention, not even a citation of other works on the same class of problems using essentially the same RG strategy but slightly different techniques or focusing on slightly different situations. The ring expansion is not the only way to organize expansions such that analyticity statements can be proven. The same is also possible using tree expansions à la Brydges-Battle-Federbush [14,15] or bounds obtained from the Brydges-Kennedy formulas [16,17]. The organization of the expansions using Laplacians in the field variables done in some of these works provides an alternative to the ring expansion technique. An introduction to the algebraic aspects, some background, as well as a detailed proof of the relation to the second-quantized Hamiltonian formulation, and a list of relevant references, is provided, e.g., in [18]. Another recent reference about the RG is [19].
To summarize, this is an excellent technical introduction for graduate students and researchers new to the field who want to start reading the series of papers [1]. To get a motivation and a larger perspective of the study of these problems, or to learn about alternative techniques, readers need to consult other references.
Publication date: 28 Apr 2004
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