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Mathematical Aspects of Quantum Field Theory
Lecture Two: Renormalization and Effective Field Theories
Timothy Nguyen
1
Renormalization
We concluded the previous lecture by discussing the finite dimensional version of Wick’s lemma,
which computes integrals of polynomials against Gaussian measures as a sum over Feynman diagrams. In today’s lecture, we will see how this procedure carries over to the infinite dimensional
setting of (perturbative) quantum field theory.
It is instructive to work with a concrete example, though all of our analysis applies to more
general theories. Consider φ4 theory in dimension d with classical action given by
Z
1
2
4
d
φ(x)(∆ + m )φ(x) + λφ (x) .
S(φ) = d x
2
We wish to evaluate the path integral
Z
Z=
Dφe−S(φ)/~
(1.1)
as a formal power series expansion in1 ~. This involves two steps. First, we regardRDφe−S(φ)/~ as the
2
d
4
)φi/2~
product of the Gaussian measure dµ := Dφe−hφ,(∆+m
and the function e d xλφ (x)/~ which
R d
4
d xλφ (x)/~
we regard
as a Taylor series in the interaction
R d as 4a perturbation. We then expand e
I = d xλφ (x) and attempt to apply Wick’s lemma term by term to the resulting polynomial
interactions.
Wick’s lemma tells us that to integrate polynomials in φ against dµ, we have to contract the
entries of the polynomial with the “inverse matrix” of the kinetic operator ∆ + m2 . More precisely,
we contract using the integral kernel of the covariance operator (∆ + m2 )−1 defining dµ. Letting
m = 0 to make formulas simpler, this integral kernel is given explicitly by
(
cd log |x − y| d = 2
P (x, y) =
(1.2)
cd |x − y|2−d d > 2,
where cd is an appropriate constant. The operator P is also known as a propagator (though this
name is more apt for Lorentzian field theories for which we have dynamics).
Consider a typical Feynman diagram arising from φ4 theory such as
1 In other contexts, ~ is set to unity and one performs an expansion in the coupling constant λ. These two expansions
can be related via the rescaling φ → ~1/2 φ which eliminates the ~ in the denominator of (1.1).
1
Figure 1: Three loop Feynman diagram in φ4 -theory.
It will typically have loops, and these fall into two types: those that begin and end on the same
vertex and those formed out of a path of edges joining distinct vertices. If we attempt to the write the
integral
to the above Feynman diagram by contracting propagators into the polynomial
R d 4 associated
R
d xφ (x) d4 yφ4 (y) in the appropriate way, we obtain
Z
dd xdd yP (x, x)P (x, y)2 P (y, y).
(1.3)
We can see why this integral is problematic. The first problem is that P (x, y) is singular along the
diagonal, so that P (x, x) is ill-defined. The second problem is that P (x, y), being a distribution, is
such that powers of P (x, y) may no longer be a distribution. Indeed, for d = 4, we have P (x, y)2 ∼
|x − y|−4 is no longer locally integrable and thus does not define a distribution.
Thus, Wick’s lemma applied naively in the infinite-dimensional setting gives us divergent integral
expressions. This fact of life is one of the primary reasons why infinite quantities plague quantum
field theory. Moreover, it is the way in which these infinite quantities are handled that gives rise to
many interesting (and difficult) features of quantum field theory.
Making sense of divergent Feynman integrals requires two procedures: regularization and renormalization. Regularization is, as the name indicates, a way of replacing divergent expressions such
as (1.3) with tamed versions that are well-defined. However, just as there is no canonical way to
smooth out a rough function, likewise there is no canonical way in which to regulate the Feynman
diagrams of a quantum field theory. Indeed, many methods of regularization have been studied each
of which have their own advantages and disadvantages. Among those that are most well-known, we
can group these methods into three types:
• propagator regularization
• analytic continuation
• lattice regularization
The first method replaces the propagator with a smoothed out version, so that the latter restricts
to the diagonal and has iterated products which remain distrbutions. The second method seeks to
analytically continue expressions involving a discrete parameter (typically the dimension d of spacetime) to those defined on the complex plane, from which Feynman integrals become well-defined away
from isolated poles. The third method replaces a continuum theory with a discrete analog defined
on a (finite) lattice, so that Feynman integrals become products of well-defined finite-dimensional
integrals.
We will proceed by using what is known as heat kernel regularization, which falls under the first
of the above regularization methods. In this approach, we replace the propagator (1.2) with the
regulated version
Z L
P (, L) =
e−t∆ dt,
(1.4)
2
where L > > 0. One should regard and L as small distance (ultraviolet) and large distance
(infrared ) regulating parameters, respectively. This is easily seen from the Fourier transform of
P (, L), which has Fourier modes with magnitude outside the range [L−1/2 , −1/2 ] heavily suppressed.
One also readily verifies that P (0, ∞) = P in the sense of distributions, so that we recover our original
propagator when the ultraviolet regulator is sent to zero and the infrared regulator is sent to infinity.
Given that the heat kernels e−t∆ yield transition probabilities for the associated diffusion equation
u˙ = −Du, heat kernel regularization also goes by the name of proper time regularization, since (1.4)
expresses P (, L)(x, x0 ) as a sum over the transition probabilities for a particle to travel from x0 to
x in time between and L.
The heat kernel approach has the advantage of being extremely flexible, allowing generalization to
theories with arbitrary kinetic operators of Laplace-type (in particular, to those defined on general
Riemannian manifolds). This is in contrast to the alternative, more frequently used momentum
cutoff procedure, in which a sharp cutoff in Fourier space is used to tame P . Observe that this latter
procedure is only well suited for constant coefficient kinetic operators.
Once we let either → 0 or L → ∞, the well-defined Feynman diagrams with propagators
P (, L) may diverge. Divergences arising from these two limits are called ultraviolet and infrared
divergences, respectively. Renormalization, roughly speaking, is a way of consistently eliminating
ultraviolet divergent expressions from regulated Feynman diagrams. This is achieved in two steps.
First, we add counterterms to our action S(φ), which are ultraviolet divergent interactions. Second,
we arrange these counterterms so that as the ultraviolet regulator is removed, the divergences of
the counterterms cancel those arising from the regulated Feynman diagrams, from which we obtain
well-defined finite integral expressions.
An oversimplified illustration P
of this procedure is to consider the the way in which one might
∞
make sense of the divergent sum n=1 n1 . One can truncate (i.e. regulate) the sum by replacing it
PN 1
with n=1 n for some finite N . Then one can add the counterterm − log N so that
!
N
X
1
lim
− log N < ∞.
N →∞
n
n=1
P∞
An alternative procedure is to consider the analytically continued expression n=1 n1s , which is just
the Riemann zeta function. The function is analytic in s and has a simple pole at s = 1 with residue
1
equal to unity. One can thus subtract the pole s−1
so that
!
∞
X
1
1
lim
−
< ∞.
s→1
ns
s−1
n=1
The (perturbative) renormalization of quantum field theories is a formally similar procedure,
though much more involved and delicate. But it is worthwhile to pause and note that, just as with
the above finite dimensional example, renormalization, like regularization, involves non-canonical
choices. Indeed, counterterms can always be shifted by terms which are ultraviolet finite (e.g. in the
above, we could have subtracted 1 + log N instead of log N ). In certain cases, such as dimensional
regularization, one might have a canonical choice such as a minimal subtraction scheme (whereby
one only subtracts poles from divergent expressions and no finite terms). But even then, that is
a choice one must make. There are times when one will want to subtract finite terms. Moreover,
outside of dimensional regularization, a general regularization scheme does not provide a canonical
choice of counterterms.
3
Figure 2: One-loop four-point function in φ4 -theory.
As we have described renormalization above, more constraints need to be placed on the nature of
the counterterms, else one could do something trivial like subtract off terms corresponding to entire
Feynman diagrams and obtain zero as the final answer. The key constraint is that counterterms are
to be given by local interactions, that is, those that are defined by polynomials in the derivatives
of the fields. Moreover, they must depend only on the ultraviolet parameter and not the infrared
parameter L. Infrared divergences have to be dealt separately from ultraviolet ones, and we will
not deal with them in this lecture series. (For all practical purposes, one always fixes an infrared
regulator and then performs renormalization, which is a purely ultraviolet phenomenon.)
Before stating our main theorem about renormalization, let us see how renormalization works in
an actual example arising from φ4 theory in dimension 4 using heat kernel regularization. Consider
the one-loop four-point interaction of Figure 2. Suppose we place the regulated propagator
1
P (, L) =
(4πt)2
Z
L
2
e−|x−y|
/4t
dt
on the internal edges and label the four external tails by the function φi , 1 ≤ i ≤ 4. Denote
the integral corresponding to such a diagram by I 1,4 [, L]. Let ' denote equality up to an overall
constant, we have
I
1,4
Z
L
[, L] '
Z
dt1
Z
=
=
L
Z
dt2
d4 xd4 y
L
Z
L
Z
1
e
t21 t22
−|x−y|2 /4t1 −|x−y|2 /4t2
e
φ1 (x)φ2 (x)φ3 (y)φ4 (y)
−1
1 −|x−y|2 1 (t−1
4 1 +t2 ) φ (x)φ (x)φ (y)φ (y)
e
1
2
3
4
2
2
t1 t2
Z L
Z L
Z
(t−1 + t−1 )−2 −|w|2 /4
dt1
dt2 d4 xd4 w 1 2 22
e
×
t1 t2
−1 −1/2
−1 −1/2
[φ1 (x)φ2 (x)φ3 (x + (t−1
w)φ4 (x + (t−1
w)]
1 + t2 )
1 + t2 )
dt1
dt2
Z
≡
dt1 dt2
[,L]2
d4 xd4 y
1
(t1 + t2 )2
Z
d4 xφ1 (x)φ2 (x)φ3 (x)φ4 (x)
Z
2
d4 we−|w|
/4
(1.5)
where ≡ denotes equivalence up to terms which are finite in the limit → 0 with L < ∞ fixed. In
deducing (1.5), observe that the → 0 singular behavior of I 1,4 [, L] occurs along the diagonal x = y
(w = 0). Indeed, Taylor expanding the φi about w = 0, only the zeroth order part φi (x) contributes
to an integral singular as → 0 (higher order Taylor coefficients contribute higher powers of the ti ,
thus taming the integral over the ti ). Here, we also assume that the φi are say compactly supported
so that the I 1,4 [, L] is a priori defined for all > 0 and that Taylor expanding along the diagonal
yields the correct → 0 asymptotics.
Computing the leading term of (1.5) and setting all the φi equal to φ, we obtain the local
4
functional
h
Z
i Z
log ( + L)/2 − log 2L/(L + )
d4 xφ4 (x) ≡ (log −1 ) d4 xφ4 (x).
As a consequence,
lim I 1,4 [, L] − c(log −1 )
→0
Z
d4 xφ4 (x)
(1.6)
exists, where c is a readily computable constant which we suppressed in the above computations.
The second term of (1.6) is thus a suitable counterterm for the I 1,4 [, L] interaction.
The above computation was quite simple as far as renormalization goes because it only involves
one loop. In a Feynman diagram with multiple loops, divergences arising from subdiagrams and
their counterterms intermingle in nontrivial ways and it is a nontrivial task to extract the overall
divergence of the diagram.
Let us formalize the renormalization procedure and then state our main theorem. Define
W (P (, L), I) to be ~−1 times the sum over all connected Feynman diagrams2 obtained from I/~
using Wick’s lemma with propagator ~P (, L). (Thus, such Feynman diagrams are automatically
weighted by the appropriate combinatorial factors.) We can make this definition more precise by
introducing some notation.
Given a real vector space V , let Hom(V ⊗n , R) denote the space of n-multilinear maps from V ⊗n
to R. For v ∈ V , define the operator
∂v : Hom(V ⊗n , R) → Hom(V ⊗n−1 , R)
X
∗
∗
v1∗ ⊗ · · · ⊗ vn∗ 7→
vi∗ (v) v1∗ ⊗ · · · ⊗ vi−1
⊗ vi+1
⊗ · · · ⊗ vn∗ .
(1.7)
i
In other words, ∂v is the directional derivative with respect to v. It extends readily to an operation
on the symmetric algebra and completed symmetric algebras
Sym(V ∗ ) = ⊕n≥0 Hom(Symn (V ), R)
Y
d ∗) =
Sym(V
Hom(Symn (V ), R),
(1.8)
(1.9)
n≥0
which are the spaces of polynomial and formal power series functions on V , respectively. More
generally, given an element of K = u ⊗ v ∈ V ⊗2 , we can define the operation
∂K =
1
∂v ∂u .
2
(1.10)
This operation extends bilinearly to any K ∈ V ⊗2 and yields a well-defined contraction operator ∂K
for any K ∈ Sym2 (V ).
The above notation is convenient, because we can use it to reformulate Wick’s lemma very
succinctly. Given a positive definite symmetric matrix A on V = Rd , let dµA denote normalized
2 On flat space Rd , one also ignores vacuum diagrams, i.e., those without external legs. This is because translation
invariance of the propagator P (x, y) = P (x − y) implies that integral expressions such as (1.3) are divergent no matter
how P (x, y) is regulated, since they are proportional to the volume of space.
5
Gaussian measure with covariance A−1 as in Lecture One. Letting Aij be the matrix entries of A
with respect to the standard orthornormal basis ei of V , let P denote the element of V ⊗ V given
by Aij ei ej , where recall Aij are the entries of the inverse matrix of Aij .
Lemma 1.1. (Wick’s Lemma) Let f (x) be any polynomial function. Then
Z
f (x)dµA = (e∂P f )(0).
(1.11)
This is a simple combinatorial exercise. Indeed, exponentiating the operator ∂P implements
the sum over Wick contractions with the appropriate combinatorial factors. In the Feynman diagrammatic picture, evaluation of the right-hand side of (1.11) at zero means we consider only those
Feynman diagrams that have no external tails. If we drop this condition so that we get a function
(e∂P f )(x), then we get a sum over all Feynman diagrams with arbitrary number of external tails,
the number of external tails being the polynomial degree in x of the monomial corresponding to the
diagram.
In the quantum field theoretic setting, the regulated propagator P (, L) is given by a smooth
symmetric integral kernel and can be regarded as an element of3 Sym2 (C ∞ (Rd )) (in the case of
the scalar field theory we have been considering). This allows us to make sense of the contraction
operation ∂~P (,L) with functionals. Thus, in light of the above lemma, e∂~P (,L) eI/~ is the sum
over all Feynman diagrams with ~P (, L) placed on edges and with interactions from I/~ placed on
vertices. It is a simple combinatorial fact that such a sum can be written as the exponential of the
sum over all Feynman diagrams that are connected4 . Thus, we have
W (P (, L), I) = log ~e∂~P (,L) eI/~ .
(1.12)
To clarify, the term log is the right inverse of the exponential function defined on (formal) power
series; intrinsically, it is the operation which maps formal power series with unital constant term to
n+1
n
P
another power series given by the formula log x = log(1 + (x − 1)) = n (−1) n(x−1) . In the above,
interactions are power series in the sense of being elements of (1.9) valued in power series in ~.
The sum of diagrams in (1.12) can be interpreted as the set of effective interactions obtained by
letting particles (which are represented by the fields of the theory) interact for proper time between
and L in the sum over histories picture of quantum field theory. Denote the expression (1.12) more
succinctly by I[, L]. We have the decomposition
X
I[, L] =
~i I i,j [, L],
i≥0,j>0
where each I (i,j) [, L] consists of those Feynman diagrams with i loops and j external tails. (It is
an easy exercise to check from the definition (1.12) that the power of ~ associated to a diagram is
precisely the number of its loops.) We will use a superscript (i, j) to denote the i-loop j-external tail
component of any interaction from now. Observe that we have the natural lexicographic ordering
on such pairs, where (i, j) < (i0 , j 0 ) if i < i0 or i = i0 , j < j 0 .
3 The tensor product which we use is in the sense of nuclear Frechet spaces, which has the property that C ∞ (X) ⊗
C ∞ (Y ) = C ∞ (X × Y ). Likewise spaces of distributions satisfy the analogous property with respect to tensor product.
4 Connected, recall, means no component of the graph is a vacuum diagram.
6
Let → 0. The terms I i,j [, L] will in general diverge, and we explicitly saw this in the case
1,4
of
[, L] for φ4 theory. The goal of renormalization is to introduce counterterms I CT () =
P Ii CT,i,j
~I
(), with each I CT,i,j a local interaction depending on the ultraviolet parameter , such
that
lim W (P (, L), I + I CT ())
→0
i,j
exists. That is, each term W of the above limit exists.
Based on the above, counterterms can be constructed inductively in the lexicographic ordering
of diagrams. This is because Feynman diagrams with a greater lexicographic order are obtained
from Wick contractions from diagrams with lower lexicographic order5 . To any fixed order in perturbation theory, the counterterms introduced up to that stage render diagrams finite only up to
that order. As one increments the order in perturbation theory, new counterterms are needed to
resolve (nested) divergences from Feynman diagrams involving the original interactions I and the
counterterms already introduced. (In the physics literature, the order in perturbation theory is usually phrased in terms of the number of loops of Feynman diagrams.) Physically, Feynman diagrams
of higher order are regarded as being less significant in perturbation theory since ~ is regarded as a
small parameter, and so for experimental purposes, one only ever renormalizes to some finite order.
We have our main theorem:
Theorem 1.2. There exists a collection of local countereterms I CT () such that lim→0 W (P (, L), I+
I CT ()) exists as a formal power series interaction.
This theorem is proven in [1] for a very general class of theories by using the asymptotic expansion
of the heat kernel at small times and a clever inductive scheme. By the former, I mean that e−tD , for
D a Laplace type operator, has a small t asymptotic expansion in terms of integral kernels localized
on the diagonal. This localization property is what allows us to extract counterterms which are local
from divergent Feynman diagrams.
While I should note that I am by no means an expert on the various methods of renormalization,
the proof of the above theorem in [1] is, in my opinion, remarkable for its elegance and succinctness. Alternative methods of renormalization, to name just two, include those involving technical
combinatorics (the BPHZ renormalization scheme [5]) or else those involving a more sophisticated
use of distribution theory (the Epstein-Glaser approach [2]). However, it is worth remarking, based
on personal experience, that despite renormalization being treated carefully in certain places, much
of the quantum field theory literature (i.e. physics textbooks and papers) handles renormalization
in a way that is extraordinarily difficult for a mathematician to sort through, much less accept.
Quite simply, the regularization and renormalization procedures are often not carried out explicitly
(to all orders in perturbation theory), or if they are, then they often used in tandem with formal
(nonrigorous) manipulations of the path integral.
Moreover, renormalization is just the beginning of the story. One is ultimately interested in
such issues as symmetries of quantum field theories, which we will come to shortly. The nature
of such symmetries in a given quantum field theory very much depend on how one regulates and
renormalizes the theory, and so the former can be dealt with rigorously only if the latter is as well. A
great feature of [1] is that it addresses the renormalization procedure in such a way that is compatible
with making sense of symmetries of a quantum field theory in a transparently rigorous fashion.
5 The
perturbing interaction I is assumed to be at least cubic (modulo ~).
7
Remark 1.3. The term renormalization has been used both in noun and verb form. For me, this
usage refers to the inductive construction of counterterms so as to render all effective interactions
finite. On the other hand, the adjective renormalizable means something much stronger. According
to one point of view, a quantum field theory is renormalizable if only finitely many parameters need
adjustment at each order of perturbation theory in order to render the theory finite. In particular,
for every i, there are only finitely many nonzero counterterms I CT,(i,j) (). One can regard this as a
statement about the predictive power of the theory in that only finitely many physical parameters
(and thus, empirical experiments) are needed to determine the theory. It is for this reason that
renormalizable quantum field theories form a distinguished class of theories.
2
Effective Field Theories and Their Symmetries
Summarizing what we have done thus far, via regularization and renormalization, we obtain a
collection of renormalized effective interactions
I[L] = lim W (P (, L), I + I CT ()).
→0
Since we have let → 0, these interactions include all quantum fluctuations that happen at length
scale below L. Counterterms, which are auxiliary quantities one cannot directly measure, eliminate
the divergences that occur as we probe to arbitrarily small lengths. Moreover, the introduction of
the parameter L means that what we actually obtain is a family of effective interactions {I[L]}L
each indexed by the length scale L.
Remark 2.1. Over a compact space, one can let L → ∞ since no infrared regulator is needed. It
is only on a noncompact space that L 6= 0 may be needed (particularly for massless theories). One
can regard the introduction of the scale L as the introduction of a mass.
The notion of effective field theories described by a range of scales is a very general one and its
interpretation depends on the particular details of the situation. In the above, L is a continuous
length parameter and we think of the interactions I[L] as providing an effective description of
phenomenon happening at lengths greater than L. If we were on a lattice, then a natural scale
would be the lattice spacing, and an effective theory can be obtained on a lattice of say double the
size by suitably averaging the individual degrees of freedom at each lattice site of a two-by-two block
into a single degree of freedom. One can iterate this procedure, thereby obtaining theories defined
on lattices that are dyadic rescalings of the fundamental one.
There is much to say about effective field theories and their role in our modern understanding
of quantum field theories after the pioneering work of K. Wilson [4]. However, in the remainder of
this lecture, we will specialize to one topic of fundamental importance, namely that of symmetry.
The notion of a symmetry for classical mechanical systems is familiar. Namely, one has a Lie
group of symmetries acting on configuration space which preserves the action. Infinitesimally, this
means that we have a Lie algebra of symmetries given by a collection of vector fields on configuration
space. Symmetries are important for a variety of reasons, since among other things, they generate
conservation laws and simplify the task of finding explicit solutions to the equations of motion.
On the other hand, we know
that in the quantum theory, what is fundamental is not the action S
R
but the path integral Z = Dφe−S(φ)/~ , where we have denoted all the fields by φ. Working purely
formally for the time being, then if we regard the integrand defining Z as an honest measure, we
8
find that it is this measure, rather than just the action, which should be preserved by a symmetry
of a quantum theory. That is, given a symmetry X of the classical theory, its manifestation as a
symmetry of the quantum theory ought to be implemented by the replacement
Classical
LX S = 0
Quantum
LX Dφe−S(φ)/~ = 0.
⇒
(2.1)
Here LX denotes the Lie derivative with respect to X.
If Dφ were a measure on a finite dimensional space, then it would be possible to make sense
of the right-hand side (2.1). However, due to the difficulty with constructing measures on infinite
dimensional spaces in quantum field theory, we content ourselves with making sense of (2.1) in
perturbation theory. Fortunately we will be able to do this, but first we reformulate the above
discussion in a different language.
There is a cohomological approach to symmetry which one can formulate purely classically but
which admits a powerful generalization to the setting of quantum field theory. Looking ahead,
what we will obtain from this approach is the Batalin-Vilkovisky (BV) formalism of quantum field
theory. In its classical formulation, suppose we are given a Lie algebra g of symmetries of a classical
field theory. Then we get an induced g-action on the space of functions of the field configurations.
Whenever we have a linear action of g on a linear space U , we can always form a chain complex,
the Chevalley-Eilenberg cochain complex C ∗ (g; U ), whose cohomology computes the Lie algebra
cohomology of g with coefficients in U . By construction, elements of the zeroth cohomology H 0 (g; U )
are those elements of U which are g-invariant.
We have thus characterized the space of elements left invariant under the action of g in cohomological terms, though in a purely tautological manner. The BV formalism goes further by (i)
including g, which in the above picture is viewed as an “external space”, into the space of fields; (ii)
adding antifields so that the total space of fields (i.e. ordinary fields and anti-fields) comes equipped
with an odd symplectic structure 6 . The latter endows the space of functionals of the fields with a
corresponding odd Poisson bracket {·, ·}.
The point of the BV construction is that a classical action and all its symmetries can be encoded
into a single master action functional S which satisfies the classical master equation {S, S} = 0.
This equation expresses the invariance of the classical action under the Lie algebra of symmetries
in question. One should regard {S, ·} as a degree one differential on the space of functions of the
fields, and the classical master equation states that S is closed with respect to this differential.
The classical master equation expresses the left-hand side of (2.1). Given a classical master
equation, one obtains a corresponding quantum master equation given by the right-hand side of
(2.1). Both these master equations have well-defined meanings when the space of all fields forms a
finite dimensional odd symplectic manifold [3].
For the Batalin-Vilkovisky formalism in the infinite dimensional setting of quantum field theory,
one starts with a well-defined classical master equation and wants to make sense of the quantum
master equation. Two features arise. First, one has to regulate the quantum master equation in a
way that is compatible with how one regulates the effective interactions that define the quantum
field theory. Second, because the quantum master equation expresses a symmetry condition while
regularization and renormalization will in general break such a symmetries, the quantum master
6 Symplectic forms in ordinary differential geometry have analogs to supermanifolds and graded manifolds. See [3]
for a quick introduction to odd symplectic geometry.
9
equation may fail to hold. In this latter situation, the quantum field theory is said to have an
anomaly.
The details of the BV formalism take time to develop and we will sketch them out in the next
lecture. Thus, in the remainder of this lecture, my goal is to give an impression using minimal
machinery of (i) how a regulated quantum master equation can be defined and (ii) how anomalies
may arise. Precise details can be found in [1].
For (ii), what I would like to emphasize is that the problem of eliminating potential anomalies,
i.e., regulating and renormalizing effective interactions to maintain the quantum master equation
order by order in ~, is a cohomological problem. Namely, we have the following informally stated
theorem:
Theorem 2.2. The space of potential obstructions to solving the quantum master equation is given
by the degree one cohomology of {S, ·} acting on the space of local functionals of the fields.
This cohomological problem arises in the following way. First, it is shown that the obstruction to
solving the quantum master equation is given by a local action functional that is {S, ·} closed. Next,
the freedom in changing a given renormalization scheme by finite counterterms allows the obstruction
to change by an arbitrary {S, ·} exact term. It follows that one can remove the obstruction by
modifying counterterms precisely if the obstruction is cohomologically trivial. The power of this
cohomological approach lies with the fact that if the first cohomology of {S, ·} is trivial, then one can
quantize without an anomaly. More generally, one might also impose enough symmetry constraints
to ensure that the obstruction vanishes cohomologically even if the obstruction space is nonzero. In
this way, what could potentially be a complicated Feynman diagramattic analysis becomes replaced
with computing Lie algebraic cohomology groups.
It remains for us to discuss how to define the quantum master equation.
2.1
Energy Scale Approach
We begin by working in a finite dimensional setting. Let W be a finite dimensional Euclidean vector
space with inner product (·, ·) and let dµ denote the associated Lebesgue measure. Let Q be a
nonpositive self-adjoint operator. We then obtain the “Gaussian” measure
dµQ = ce(φ,Qφ)/2~ dµ(φ),
where c is a constant chosen so that dµQ is a normalized Gaussian measure when restricted to the
span of the eigenspaces on which Q is nondegenerate.
Let Wλ denote the λ eigenspace of −Q. Given Λ ≥ 0, we introduce the following objects:
WΛ> = ⊕λ>Λ Wλ
WΛ≤ = ⊕λ≤Λ Wλ
dµ>
Q,Λ = dµQ |WΛ>
dµ≤
Q,Λ = dµQ |W ≤
Λ
For brevity, we also write dµQ,Λ for dµ≤
Q,Λ .
10
Let φaλ denote an orthonormal eigenbasis for Q, where −Qφaλ = λφaλ . Given 0 ≤ Λ1 < Λ2 , we
have the associated propagator
X
P (Λ1 , Λ2 ) =
Λ1 <λ<Λ2
1 a a
φ (φ , ·),
λ λ λ
which serves as an approximate inverse to −Q. In fact, it is the inverse of −Q when restricted to
⊕Λ1 <λ<Λ2 Wλ .
Let V be a polynomial vector field on W , i.e., V ∈ Sym(W ∗ ) ⊗ W and let LV denote the Lie
derivative with respect to V .
Lemma 2.3. We have the identity
Z
WΛ>
LV dµQ = LV [Λ] dµQ,Λ
(2.2)
where V [Λ] = e~∂P (Λ,∞) V |W ≤ is a vector field on WΛ≤ .
Λ
Proof. On top-forms, we have that LV = dιV , where ιV denotes contraction with the vector
field V and d is exterior derivative. Write d = d> + d≤ to denote the components of d belonging to
the WΛ> and WΛ≤ directions, respectively. Then
Z
Z
LV dµQ =
(d> + d≤ )ιV dµQ
WΛ>
WΛ>
Z
=
WΛ>
d≤ ιV (dµ>
Q,Λ dµQ,Λ )
Z
= d≤
WΛ>
dµ>
Q,Λ (ιV dµQ,Λ )
= d≤ ιV [Λ] dµQ,Λ .
In the second line, the term arising from d> vanishes since it is exact. Likewise, in passing to line
three, the only nonzero contribution to the integral arises from when V contracts with the dµQ,Λ
factor. Finally, the last line is an application of Wick’s lemma. Let QV denote LV (·, Q·)/2. Then what the above shows is that
QV
+ divV dµQ = 0
LV dµQ =
~
implies
LV [Λ] dµΛ
Q
=
QV [Λ]
+ divΛ V [Λ] dµΛ
Q =0
~
where div and divΛ are the divergence operators with respect to the Lebesgue measures dµ and
dµ|W ≤ , respectively. In other words, the invariance of the measure dµQ with respect to the vector
Λ
field V descends to an invariance of the measure dµΛ
Q with respect to the scale Λ vector field V [Λ].
11
We can generalize the above situation to formal (i.e. defined perturbatively) non-Gaussian measures as follows. Observe that LV (f dµQ ) = Lf V dµQ for any function f . In particular, if we wish
∗
d
to consider an interaction term f = eI/~ , with I ∈ Sym(W
), we can repeat the preceding analysis
I/~
but with V replaced with e V . One can verify that
(f V )[Λ] = eI[Λ]/~ VI [Λ]
(2.3)
∗
d
where I[Λ] ∈ Sym(W
)[[~]] is given by
e~∂P (Λ,∞) eI/~ |W ≤ =: eI[Λ]/~
(2.4)
Λ
and
VI [Λ] = e−I[Λ]/~ e~∂P (Λ,∞) (eI/~ V )
is the sum of all connected Feynman diagrams such that exactly one vertex is labelled by V and the
rest are with I/~. Thus,
QV
LV I
I/~
+
+ divV dµQ = 0
LV e dµQ =
~
~
implies
QV [Λ] L
VI [Λ] I[Λ]
I
+
+
div
V
[Λ]
dµΛ
LVI [Λ] eI[Λ]/~ dµΛ
=
Λ I
Q
Q = 0.
~
~
(2.5)
This latter expresses the invariance of the measure eI/~ dµQ with respect to the vector field V in
terms of objects defined at scale Λ, namely, the scale Λ effective interactions I[V ] and the scale Λ
vector field V [Λ]. In other words, a symmetry for the full theory descends to a scale Λ symmetry
for the effective theory defined at scale Λ. This equation is precisely the scale Λ quantum master
equation for this simple example.
2.2
Length Scale Approach
The previous approach to defining a regulated quantum master equation was based on energy,
namely, using a momentum cutoff to define the propagator. This quantum master equation involved
taking the Lie derivative of a scale Λ measure with respect to a scale Λ vector field.
We want to define an analogous quantum master equation in the quantum field theoretic setting,
or more precisely, an equation involving our effective interactions I[L]. Since these interactions
were defined using the length scale L introduced via heat kernel regularization, we want to define a
(length) scale L quantum master equation. We will define it by formal analogy with the quantum
master equation based on the energy scale Λ, and it expresses a symmetry of the effective interactions
I[L] in terms of a scale L vector field V [L].
Because of the infinite dimensional nature of our present situation, let us define the objects which
we employ more carefully. It will be helpful to think more abstractly, since explicit notation makes
it hard to keep track of the spaces in which objects live. To keep matters simple, we make all our
fields purely bosonic so that we can ignore sign issues arising from fermionic fields7 . Let E be a
7 Though as a consequence, much of the homological algebra is lost since the differential arising in the BV formalism
requires odd fields.
12
Euclidean vector bundle over our base space Rd and let E = Γ(E) be its space of sections. The space
E will be our space of (ordinary) fields. Power series interactions are elements of the symmetric
d ∗ ) of the dual space E ∗ of E. In general, an element of (E ∗ )⊗n is a distributional
algebra8 Sym(E
section of (E ∗ )n , where E ∗ is the dual bundle of E and denotes the external tensor product of
vector bundles. A local interaction is one supported on the small diagonal and given by integrating
a density formed pointwise out of the derivatives of the fields:
Z
I(φ) =
dd xI(φ(x), ∇φ(x), ∇2 φ(x), ...),
φ ∈ E.
Rd
In the above, I is polynomial function or more generally a power series.
d ∗ ) ⊗ E. It is thus a φ-dependent section
A (power series) vector field V is an element of Sym(E
of E. A local vector field is one in which the value of such a section at each point x depends only
upon the derivatives of φ at x. In physics notation, this is often written as
Z
∂
,
a = 1, . . . , rank(E).
V (φ) = dd xV a (φ(x), ∇φ(x), ∇2 φ(x), . . .) a
∂φ (x)
The above integration is only formal notation: if we interpret the ∂φa∂(x) as a basis of the fiber Ex ,
then the coefficients of V (φ) at x are given by the power series V a and the integration denotes that
this relation holds for all x.
Given a kinetic operator Q = −∆ : Γ(E) → Γ(E) that is a second order differential operator
which is symmetric with negative symbol, we get a regulated propagator
Z L
P (, L) =
e−t∆ dt
which is given by a smooth integral kernel. That is
P (, L) ∈ Γ (E (E ∗ ⊗ Dens))
where Dens is the bundle of densities on Rd . Since we have the canonical density dd x on Rd and the
isomorphism E ∗ ∼
= E induced from the inner product on E, we can regard9
P (, L) ∈ Γ(E E) = E ⊗ E.
In fact, P (, L) ∈ Sym2 (E ⊗ E) by symmetry of ∆.
Using the definition (1.10), which carries over to the setting of infinite dimensional vector spaces,
since P (, L) ∈ E ⊗ E, we can form the contraction ∂P (,L) with interactions and vector fields. In this
way, we have the renormalized effective interactions I[L] (as before) as well as renormalized vector
fields
CT
VI [L] = e−I[L]/~ lim e~∂P (,L) (eI+I ()/~ (V + V CT ()),
→0
which in general requires counterterms V CT () for the vertex V in addition to those arising from I
in Feynman diagrams).
8 As
before, all tensor products are in the sense of nuclear Fr´
echet spaces.
tensor product which appears is in the sense of nuclear Frechet spaces, and it allows one to write C ∞ (X) ⊗
C ∞ (Y ) = C ∞ (X × Y ) where X and Y are smooth manifolds. Generalizing to sections of bundles, we obtain
Γ(E) ⊗ Γ(E) = Γ(E E).
9 The
13
We can now write the scale L quantum master equation:
(Q + divL )(eI[L]/~ VI [L]) = 0
(2.6)
where divL is the scale L divergence operator defined as follows.
Unlike the energy scale approach in finite dimensions, it is not the case that we have a family
of measures indexed by L from which we compute the divergence. Rather, we make the following
observation. On a finite dimensional Euclidean space, the divergence operator acting on a power
∗
d
series vector field V ∈ Sym(W
) ⊗ W is given by the contraction of V with the identity tensor
∗
idW ∈ W ⊗ W . In the setting of quantum field theory, the identity operator does not have a
smooth integral kernel (it is the Dirac delta distribution). This suggests that a regulated divergence
operator is a regulated identity operator. The heat kernels e−L∆ , L > 0, provide a natural class of
such regulated identity operators.
Let KL be the integral kernel of e−L∆ symmetrized in the following way. Consider the bundle
˜ = E ⊕ (E ∗ ⊗ Dens).
E
Observe that the natural map of vector spaces
U ⊗ V → Sym2 (U ⊕ V )
u ⊗ v 7→ u ⊗ v + v ⊗ u,
induces a natural map
˜
Γ E ⊗ (E ∗ ⊗ Dens) → Γ(Sym2 (E)).
Thus, we can regard
˜
KL ∈ Γ(Sym2 (E)).
(2.7)
We have the pairing
Γ(E ∗ ⊗ Dens) × Γ(E) → R
∗
(2.8)
10
given by pointwise pairing E with E and then integrating the resulting density . This defines for
us how to contract an element of Γ(E ∗ ⊗ Dens) with an element of E. In this way, an element of
˜ can be contracted into an element of E or E ∗ , where one uses either the pairing (2.8) if
v ∈ Γ(E)
v ∈ Γ(E ∗ ⊗ Dens) or else the natural evaluation pairing E E ∗ → R if v ∈ E.
The above discussion and (2.7) implies we have a well-defined scale L divergence operator:
d ∗ ) ⊗ E → Sym(E
d ∗ ).
divL := ∂KL : Sym(E
It maps vector fields to functionals (interactions), as a divergence operator should. The scale L
quantum master equation (2.6) is sensible as a definition because of the following lemma:
Lemma 2.4. For all L, L0 > 0, the scale L quantum master equation holds if and only if the scale
L0 quantum master equation holds.
Because of this, obstruction analysis for the quantum master equation can be performed at any
scale L. In particular, we can let L → 0. The localization of the heat kernel at small times implies
that the effective interactions I[L] become more and more local. This can be used to show that
in the limit L → 0, the obstruction to solving the quantum master equation is given by a local
functional. This is one of the key steps in proving Theorem 2.2.
10 We assume that the resulting density is integrable. This will always be the case if the sections of Γ(E) are taken
to be compactly supported. Since all our interactions can be interpreted as Feynman diagrams, this corresponds to
compactly supported functions being placed on the external tails of such diagrams.
14
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(1993)
[4] K. Wilson. The renormalization group: Critical phenomenon and the Kondo problem. Rev.
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