In theoretical physics, the renormalization
group (RG) refers to a mathematical apparatus
that allows systematic investigation of the
changes of a physical system as viewed at
different scales. In particle physics, it
reflects the changes in the underlying force
laws (codified in a quantum field theory)
as the energy scale at which physical processes
occur varies, energy/momentum and resolution
distance scales being effectively conjugate
under the uncertainty principle (cf. Compton
wavelength).
A change in scale is called a scale transformation.
The renormalization group is intimately related
to scale invariance and conformal invariance,
symmetries in which a system appears the same
at all scales (so-called self-similarity).As
the scale varies, it is as if one is changing
the magnifying power of a notional microscope
viewing the system. In so-called renormalizable
theories, the system at one scale will generally
be seen to consist of self-similar copies
of itself when viewed at a smaller scale,
with different parameters describing the components
of the system. The components, or fundamental
variables, may relate to atoms, elementary
particles, atomic spins, etc. The parameters
of the theory typically describe the interactions
of the components. These may be variable couplings
which measure the strength of various forces,
or mass parameters themselves. The components
themselves may appear to be composed of more
of the self-same components as one goes to
shorter distances.
For example, in quantum electrodynamics (QED),
an electron appears to be composed of electrons,
positrons (anti-electrons) and photons, as
one views it at higher resolution, at very
short distances. The electron at such short
distances has a slightly different electric
charge than does the dressed electron seen
at large distances, and this change, or running,
in the value of the electric charge is determined
by the renormalization group equation.
== History ==
The idea of scale transformations and scale
invariance is old in physics. Scaling arguments
were commonplace for the Pythagorean school,
Euclid and up to Galileo. They became popular
again at the end of the 19th century, perhaps
the first example being the idea of enhanced
viscosity of Osborne Reynolds, as a way to
explain turbulence.
The renormalization group was initially devised
in particle physics, but nowadays its applications
extend to solid-state physics, fluid mechanics,
physical cosmology and even nanotechnology.
An early article by Ernst Stueckelberg and
André Petermann in 1953 anticipates the idea
in quantum field theory. Stueckelberg and
Petermann opened the field conceptually. They
noted that renormalization exhibits a group
of transformations which transfer quantities
from the bare terms to the counter terms.
They introduced a function h(e) in quantum
electrodynamics (QED), which is now called
the beta function (see below).
Murray Gell-Mann and Francis E. Low in 1954
restricted the idea to scale transformations
in QED, which are the most physically significant,
and focused on asymptotic forms of the photon
propagator at high energies. They determined
the variation of the electromagnetic coupling
in QED, by appreciating the simplicity of
the scaling structure of that theory. They
thus discovered that the coupling parameter
g(μ) at the energy scale μ is effectively
given by the group equation
for some function G (unspecified—nowadays
called Wegner's scaling function) and a constant
d, in terms of the coupling g(M) at a reference
scale M.
Gell-Mann and Low realized in these results
that the effective scale can be arbitrarily
taken as μ, and can vary to define the theory
at any other scale:
The gist of the RG is this group property:
as the scale μ varies, the theory presents
a self-similar replica of itself, and any
scale can be accessed similarly from any other
scale, by group action, a formal transitive
conjugacy of couplings in the mathematical
sense (Schröder's equation).
On the basis of this (finite) group equation
and its scaling property, Gell-Mann and Low
could then focus on infinitesimal transformations,
and invented a computational method based
on a mathematical flow function ψ(g) = G
d/(∂G/∂g) of the coupling parameter g,
which they introduced. Like the function h(e)
of Stueckelberg and Petermann, their function
determines the differential change of the
coupling g(μ) with respect to a small change
in energy scale μ through a differential
equation, the renormalization group equation:
The modern name is also indicated, the beta
function, introduced by C. Callan and K. Symanzik
in 1970. Since it is a mere function of g,
integration in g of a perturbative estimate
of it permits specification of the renormalization
trajectory of the coupling, that is, its variation
with energy, effectively the function G in
this perturbative approximation. The renormalization
group prediction (cf. Stueckelberg–Petermann
and Gell-Mann–Low works) was confirmed 40
years later at the LEP accelerator experiments:
the fine structure "constant" of QED was measured
to be about 1/127 at energies close to 200
GeV, as opposed to the standard low-energy
physics value of 1/137. (Early applications
to quantum electrodynamics are discussed in
the influential book of Nikolay Bogolyubov
and Dmitry Shirkov in 1959.)
The renormalization group emerges from the
renormalization of the quantum field variables,
which normally has to address the problem
of infinities in a quantum field theory (although
the RG exists independently of the infinities).
This problem of systematically handling the
infinities of quantum field theory to obtain
finite physical quantities was solved for
QED by Richard Feynman, Julian Schwinger and
Shin'ichirō Tomonaga, who received the 1965
Nobel prize for these contributions. They
effectively devised the theory of mass and
charge renormalization, in which the infinity
in the momentum scale is cut off by an ultra-large
regulator, Λ (which could ultimately be taken
to be infinite — infinities reflect the
pileup of contributions from an infinity of
degrees of freedom at infinitely high energy
scales.). The dependence of physical quantities,
such as the electric charge or electron mass,
on the scale Λ is hidden, effectively swapped
for the longer-distance scales at which the
physical quantities are measured, and, as
a result, all observable quantities end up
being finite instead, even for an infinite
Λ. Gell-Mann and Low thus realized in these
results that, while, infinitesimally, a tiny
change in g is provided by the above RG equation
given ψ(g), the self-similarity is expressed
by the fact that ψ(g) depends explicitly
only upon the parameter(s) of the theory,
and not upon the scale μ. Consequently, the
above renormalization group equation may be
solved for (G and thus) g(μ).
A deeper understanding of the physical meaning
and generalization of the renormalization
process, which goes beyond the dilation group
of conventional renormalizable theories, considers
methods where widely different scales of lengths
appear simultaneously. It came from condensed
matter physics: Leo P. Kadanoff's paper in
1966 proposed the "block-spin" renormalization
group. The blocking idea is a way to define
the components of the theory at large distances
as aggregates of components at shorter distances.
This approach covered the conceptual point
and was given full computational substance
in the extensive important contributions of
Kenneth Wilson. The power of Wilson's ideas
was demonstrated by a constructive iterative
renormalization solution of a long-standing
problem, the Kondo problem, in 1975, as well
as the preceding seminal developments of his
new method in the theory of second-order phase
transitions and critical phenomena in 1971.
He was awarded the Nobel prize for these decisive
contributions in 1982.Meanwhile, the RG in
particle physics had been reformulated in
more practical terms by C. G. Callan and K.
Symanzik in 1970. The above beta function,
which describes the "running of the coupling"
parameter with scale, was also found to amount
to the "canonical trace anomaly", which represents
the quantum-mechanical breaking of scale (dilation)
symmetry in a field theory. (Remarkably, quantum
mechanics itself can induce mass through the
trace anomaly and the running coupling.) Applications
of the RG to particle physics exploded in
number in the 1970s with the establishment
of the Standard Model.
In 1973, it was discovered that a theory of
interacting colored quarks, called quantum
chromodynamics, had a negative beta function.
This means that an initial high-energy value
of the coupling will eventuate a special value
of μ at which the coupling blows up (diverges).
This special value is the scale of the strong
interactions, μ = ΛQCD and occurs at about
200 MeV. Conversely, the coupling becomes
weak at very high energies (asymptotic freedom),
and the quarks become observable as point-like
particles, in deep inelastic scattering, as
anticipated by Feynman-Bjorken scaling. QCD
was thereby established as the quantum field
theory controlling the strong interactions
of particles.
Momentum space RG also became a highly developed
tool in solid state physics, but its success
was hindered by the extensive use of perturbation
theory, which prevented the theory from reaching
success in strongly correlated systems. In
order to study these strongly correlated systems,
variational approaches are a better alternative.
The conformal symmetry is associated with
the vanishing of the beta function. This can
occur naturally if a coupling constant is
attracted, by running, toward a fixed point
at which β(g) = 0. In QCD, the fixed point
occurs at short distances where g → 0 and
is called a (trivial) ultraviolet fixed point.
For heavy quarks, such as the top quark, it
is calculated that the coupling to the mass-giving
Higgs boson runs toward a fixed non-zero (non-trivial)
infrared fixed point.
In string theory conformal invariance of the
string world-sheet is a fundamental symmetry:
β = 0 is a requirement. Here, β is a function
of the geometry of the space-time in which
the string moves. This determines the space-time
dimensionality of the string theory and enforces
Einstein's equations of general relativity
on the geometry. The RG is of fundamental
importance to string theory and theories of
grand unification.
It is also the modern key idea underlying
critical phenomena in condensed matter physics.
Indeed, the RG has become one of the most
important tools of modern physics. It is often
used in combination with the Monte Carlo method.
== Block spin ==
This section introduces pedagogically a picture
of RG which may be easiest to grasp: the block
spin RG, devised by Leo P. Kadanoff in 1966.Consider
a 2D solid, a set of atoms in a perfect square
array, as depicted in the figure.
Assume that atoms interact among themselves
only with their nearest neighbours, and that
the system is at a given temperature T. The
strength of their interaction is quantified
by a certain coupling J. The physics of the
system will be described by a certain formula,
say the hamiltonian H(T,J).
Now proceed to divide the solid into blocks
of 2×2 squares; we attempt to describe the
system in terms of block variables, i.e.,
variables which describe the average behavior
of the block. Further assume that, by some
lucky coincidence, the physics of block variables
is described by a formula of the same kind,
but with different values for T and J : H(T',J').
(This isn't exactly true, in general, but
it is often a good first approximation.)
Perhaps, the initial problem was too hard
to solve, since there were too many atoms.
Now, in the renormalized problem we have only
one fourth of them. But why stop now? Another
iteration of the same kind leads to H(T",J"),
and only one sixteenth of the atoms. We are
increasing the observation scale with each
RG step.
Of course, the best idea is to iterate until
there is only one very big block. Since the
number of atoms in any real sample of material
is very large, this is more or less equivalent
to finding the long range behaviour of the
RG transformation which took (T,J) → (T',J')
and (T',J')→ (T",J"). Often, when iterated
many times, this RG transformation leads to
a certain number of fixed points.
To be more concrete, consider a magnetic system
(e.g., the Ising model), in which the J coupling
denotes the trend of neighbour spins to be
parallel. The configuration of the system
is the result of the tradeoff between the
ordering J term and the disordering effect
of temperature.
For many models of this kind there are three
fixed points:
T=0 and J → ∞. This means that, at the
largest size, temperature becomes unimportant,
i.e., the disordering factor vanishes. Thus,
in large scales, the system appears to be
ordered. We are in a ferromagnetic phase.
T → ∞ and J → 0. Exactly the opposite;
here, temperature dominates, and the system
is disordered at large scales.
A nontrivial point between them, T = Tc and
J = Jc. In this point, changing the scale
does not change the physics, because the system
is in a fractal state. It corresponds to the
Curie phase transition, and is also called
a critical point.So, if we are given a certain
material with given values of T and J, all
we have to do in order to find out the large-scale
behaviour of the system is to iterate the
pair until we find the corresponding fixed
point.
== Elementary theory ==
In more technical terms, let us assume that
we have a theory described by a certain function
Z
{\displaystyle Z}
of the state variables
{
s
i
}
{\displaystyle \{s_{i}\}}
and a certain set of coupling constants
{
J
k
}
{\displaystyle \{J_{k}\}}
. This function may be a partition function,
an action, a Hamiltonian, etc. It must contain
the whole description of the physics of the
system.
Now we consider a certain blocking transformation
of the state variables
{
s
i
}
→
{
s
~
i
}
{\displaystyle \{s_{i}\}\to \{{\tilde {s}}_{i}\}}
, the number of
s
~
i
{\displaystyle {\tilde {s}}_{i}}
must be lower than the number of
s
i
{\displaystyle s_{i}}
. Now let us try to rewrite the
Z
{\displaystyle Z}
function only in terms of the
s
~
i
{\displaystyle {\tilde {s}}_{i}}
. If this is achievable by a certain change
in the parameters,
{
J
k
}
→
{
J
~
k
}
{\displaystyle \{J_{k}\}\to \{{\tilde {J}}_{k}\}}
, then the theory is said to be renormalizable.
For some reason, most fundamental theories
of physics such as quantum electrodynamics,
quantum chromodynamics and electro-weak interaction,
but not gravity, are exactly renormalizable.
Also, most theories in condensed matter physics
are approximately renormalizable, from superconductivity
to fluid turbulence.
The change in the parameters is implemented
by a certain beta function:
{
J
~
k
}
=
β
(
{
J
k
}
)
{\displaystyle \{{\tilde {J}}_{k}\}=\beta
(\{J_{k}\})}
, which is said to induce a renormalization
flow (or RG flow) on the
J
{\displaystyle J}
-space. The values of
J
{\displaystyle J}
under the flow are called running couplings.
As was stated in the previous section, the
most important information in the RG flow
are its fixed points. The possible macroscopic
states of the system, at a large scale, are
given by this set of fixed points.If these
fixed points correspond to a free field theory,
the theory is said to exhibit quantum triviality,
possessing what is called a Landau pole, as
in quantum electrodynamics. For a φ4 interaction,
Michael Aizenman proved that this theory is
indeed trivial, for space-time dimension D
≥ 5.For D = 4, the triviality has yet to
be proven rigorously, but lattice computations
have provided strong evidence for this. This
fact is important as quantum triviality can
be used to bound or even predict parameters
such as the Higgs boson mass in asymptotic
safety scenarios. Numerous fixed points appear
in the study of lattice Higgs theories, but
the nature of the quantum field theories associated
with these remains an open question.Since
the RG transformations in such systems are
lossy (i.e.: the number of variables decreases
- see as an example in a different context,
Lossy data compression), there need not be
an inverse for a given RG transformation.
Thus, in such lossy systems, the renormalization
group is, in fact, a semigroup.
== Relevant & irrelevant operators and universality
classes ==
Consider a certain observable A of a physical
system undergoing an RG transformation. The
magnitude of the observable as the length
scale of the system goes from small to large
may be: (a) always increasing, (b) always
decreasing or (c) other. In the first case,
the observable is said to be a relevant observable;
in the second, irrelevant and in the third,
marginal.
A relevant observable is needed to describe
the macroscopic behaviour of the system; an
irrelevant observable is not. Marginal observables
may or may not need be taken into account.
A remarkable broad fact is that most observables
are irrelevant, i.e., the macroscopic physics
is dominated by only a few observables in
most systems. As an example, in microscopic
physics, to describe a system consisting of
a mole of carbon-12 atoms we need of the order
of 1023 (Avogadro's number) variables, while
to describe it as a macroscopic system (12
grams of carbon-12) we only need a few.
Before Wilson's RG approach, there was an
astonishing empirical fact to explain: the
coincidence of the critical exponents (i.e.,
the exponents of the reduced-temperature dependence
of several quantities near a second order
phase transition) in very disparate phenomena,
such as magnetic systems, superfluid transition
(Lambda transition), alloy physics, etc. Thus,
in general, thermodynamic features of a system
near a phase transition depend only on a small
number of variables, such as dimensionality
and symmetry, but are insensitive to details
of the underlying microscopic properties of
the system.
This coincidence of critical exponents for
ostensibly quite different physical systems
is called universality−−and is now successfully
explained by the RG: essentially by showing
that the differences among all such phenomena
are, in fact, traceable to such irrelevant
observables, while the relevant observables
are shared in common.
Thus, many macroscopic phenomena may be grouped
into a small set of universality classes,
specified by the shared sets of relevant observables.
== Momentum space ==
Renormalization groups, in practice, come
in two main "flavours". The Kadanoff picture
explained above refers mainly to the so-called
real-space RG.
Momentum-space RG on the other hand, has a
longer history despite its relative subtlety.
It can be used for systems where the degrees
of freedom can be cast in terms of the Fourier
modes of a given field. The RG transformation
proceeds by integrating out a certain set
of high-momentum (large-wavenumber) modes.
Since large wavenumbers are related to short-length
scales, the momentum-space RG results in an
essentially analogous coarse-graining effect
as with real-space RG.
Momentum-space RG is usually performed on
a perturbation expansion. The validity of
such an expansion is predicated upon the actual
physics of a system being close to that of
a free field system. In this case, one may
calculate observables by summing the leading
terms in the expansion.
This approach has proved successful for many
theories, including most of particle physics,
but fails for systems whose physics is very
far from any free system, i.e., systems with
strong correlations.
As an example of the physical meaning of RG
in particle physics, consider an overview
of charge renormalization in quantum electrodynamics
(QED). Suppose we have a point positive charge
of a certain true (or bare) magnitude. The
electromagnetic field around it has a certain
energy, and thus may produce some pairs of
(e.g.) electrons-positrons, which will be
annihilated very quickly. But, in their short
life, the electron will be attracted by the
charge, and the positron will be repelled.
Since this happens continuously, these pairs
are effectively screening the charge from
abroad. Thus, the measured strength of the
charge will depend on how close to our probes
it may enter. Hence a dependence of a certain
coupling constant (here, the electric charge)
with distance scale.
Momentum and length scales are related inversely,
according to the de Broglie relation: the
higher the energy or momentum scale we may
reach, the lower the length scale we may probe
and resolve. Therefore, the momentum-space
RG practitioners sometimes declaim to integrate
out high momenta or high energy from their
theories.
== Exact renormalization group equations ==
An exact renormalization group equation (ERGE)
is one that takes irrelevant couplings into
account. There are several formulations.
The Wilson ERGE is the simplest conceptually,
but is practically impossible to implement.
Fourier transform into momentum space after
Wick rotating into Euclidean space. Insist
upon a hard momentum cutoff, p2 ≤ Λ2 so
that the only degrees of freedom are those
with momenta less than Λ. The partition function
is
Z
=
∫
p
2
≤
Λ
2
D
ϕ
exp
⁡
[
−
S
Λ
[
ϕ
]
]
.
{\displaystyle Z=\int _{p^{2}\leq \Lambda
^{2}}{\mathcal {D}}\phi \exp \left[-S_{\Lambda
}[\phi ]\right].}
For any positive Λ' less than Λ, define
SΛ' (a functional over field configurations
φ whose Fourier transform has momentum support
within p2 ≤ Λ' 2) as
exp
⁡
(
−
S
Λ
′
[
ϕ
]
)
=
d
e
f
∫
Λ
′
≤
p
≤
Λ
D
ϕ
exp
⁡
[
−
S
Λ
[
ϕ
]
]
.
{\displaystyle \exp \left(-S_{\Lambda '}[\phi
]\right)\ {\stackrel {\mathrm {def} }{=}}\ \int
_{\Lambda '\leq p\leq \Lambda }{\mathcal {D}}\phi
\exp \left[-S_{\Lambda }[\phi ]\right].}
Obviously,
Z
=
∫
p
2
≤
Λ
′
2
D
ϕ
exp
⁡
[
−
S
Λ
′
[
ϕ
]
]
.
{\displaystyle Z=\int _{p^{2}\leq {\Lambda
'}^{2}}{\mathcal {D}}\phi \exp \left[-S_{\Lambda
'}[\phi ]\right].}
In fact, this transformation is transitive.
If you compute SΛ′ from SΛ and then compute
SΛ″ from SΛ′, this gives you the same
Wilsonian action as computing SΛ″ directly
from SΛ.
The Polchinski ERGE involves a smooth UV regulator
cutoff. Basically, the idea is an improvement
over the Wilson ERGE. Instead of a sharp momentum
cutoff, it uses a smooth cutoff. Essentially,
we suppress contributions from momenta greater
than Λ heavily. The smoothness of the cutoff,
however, allows us to derive a functional
differential equation in the cutoff scale
Λ. As in Wilson's approach, we have a different
action functional for each cutoff energy scale
Λ. Each of these actions are supposed to
describe exactly the same model which means
that their partition functionals have to match
exactly.
In other words, (for a real scalar field;
generalizations to other fields are obvious),
Z
Λ
[
J
]
=
∫
D
ϕ
exp
⁡
(
−
S
Λ
[
ϕ
]
+
J
⋅
ϕ
)
=
∫
D
ϕ
exp
⁡
(
−
1
2
ϕ
⋅
R
Λ
⋅
ϕ
−
S
int
Λ
[
ϕ
]
+
J
⋅
ϕ
)
{\displaystyle Z_{\Lambda }[J]=\int {\mathcal
{D}}\phi \exp \left(-S_{\Lambda }[\phi ]+J\cdot
\phi \right)=\int {\mathcal {D}}\phi \exp
\left(-{\tfrac {1}{2}}\phi \cdot R_{\Lambda
}\cdot \phi -S_{{\text{int}}\,\Lambda }[\phi
]+J\cdot \phi \right)}
and ZΛ is really independent of Λ! We have
used the condensed deWitt notation here. We
have also split the bare action SΛ into a
quadratic kinetic part and an interacting
part Sint Λ. This split most certainly isn't
clean. The "interacting" part can very well
also contain quadratic kinetic terms. In fact,
if there is any wave function renormalization,
it most certainly will. This can be somewhat
reduced by introducing field rescalings. RΛ
is a function of the momentum p and the second
term in the exponent is
1
2
∫
d
d
p
(
2
π
)
d
ϕ
~
∗
(
p
)
R
Λ
(
p
)
ϕ
~
(
p
)
{\displaystyle {\frac {1}{2}}\int {\frac {d^{d}p}{(2\pi
)^{d}}}{\tilde {\phi }}^{*}(p)R_{\Lambda }(p){\tilde
{\phi }}(p)}
when expanded.
When
p
≪
Λ
{\displaystyle p\ll \Lambda }
, RΛ(p)/p2 is essentially 1. When
p
≫
Λ
{\displaystyle p\gg \Lambda }
, RΛ(p)/p2 becomes very very huge and approaches
infinity. RΛ(p)/p2 is always greater than
or equal to 1 and is smooth. Basically, this
leaves the fluctuations with momenta less
than the cutoff Λ unaffected but heavily
suppresses contributions from fluctuations
with momenta greater than the cutoff. This
is obviously a huge improvement over Wilson.
The condition that
d
d
Λ
Z
Λ
=
0
{\displaystyle {\frac {d}{d\Lambda }}Z_{\Lambda
}=0}
can be satisfied by (but not only by)
d
d
Λ
S
int
Λ
=
1
2
δ
S
int
Λ
δ
ϕ
⋅
(
d
d
Λ
R
Λ
−
1
)
⋅
δ
S
int
Λ
δ
ϕ
−
1
2
Tr
⁡
[
δ
2
S
int
Λ
δ
ϕ
δ
ϕ
⋅
R
Λ
−
1
]
.
{\displaystyle {\frac {d}{d\Lambda }}S_{{\text{int}}\,\Lambda
}={\frac {1}{2}}{\frac {\delta S_{{\text{int}}\,\Lambda
}}{\delta \phi }}\cdot \left({\frac {d}{d\Lambda
}}R_{\Lambda }^{-1}\right)\cdot {\frac {\delta
S_{{\text{int}}\,\Lambda }}{\delta \phi }}-{\frac
{1}{2}}\operatorname {Tr} \left[{\frac {\delta
^{2}S_{{\text{int}}\,\Lambda }}{\delta \phi
\,\delta \phi }}\cdot R_{\Lambda }^{-1}\right].}
Jacques Distler claimed without proof that
this ERGE is not correct nonperturbatively.The
effective average action ERGE involves a smooth
IR regulator cutoff.
The idea is to take all fluctuations right
up to an IR scale k into account. The effective
average action will be accurate for fluctuations
with momenta larger than k. As the parameter
k is lowered, the effective average action
approaches the effective action which includes
all quantum and classical fluctuations. In
contrast, for large k the effective average
action is close to the "bare action". So,
the effective average action interpolates
between the "bare action" and the effective
action.
For a real scalar field, one adds an IR cutoff
1
2
∫
d
d
p
(
2
π
)
d
ϕ
~
∗
(
p
)
R
k
(
p
)
ϕ
~
(
p
)
{\displaystyle {\frac {1}{2}}\int {\frac {d^{d}p}{(2\pi
)^{d}}}{\tilde {\phi }}^{*}(p)R_{k}(p){\tilde
{\phi }}(p)}
to the action S, where Rk is a function of
both k and p such that for
p
≫
k
{\displaystyle p\gg k}
, Rk(p) is very tiny and approaches 0 and
for
p
≪
k
{\displaystyle p\ll k}
,
R
k
(
p
)
≳
k
2
{\displaystyle R_{k}(p)\gtrsim k^{2}}
. Rk is both smooth and nonnegative. Its large
value for small momenta leads to a suppression
of their contribution to the partition function
which is effectively the same thing as neglecting
large-scale fluctuations.
One can use the condensed deWitt notation
1
2
ϕ
⋅
R
k
⋅
ϕ
{\displaystyle {\frac {1}{2}}\phi \cdot R_{k}\cdot
\phi }
for this IR regulator.
So,
exp
⁡
(
W
k
[
J
]
)
=
Z
k
[
J
]
=
∫
D
ϕ
exp
⁡
(
−
S
[
ϕ
]
−
1
2
ϕ
⋅
R
k
⋅
ϕ
+
J
⋅
ϕ
)
{\displaystyle \exp \left(W_{k}[J]\right)=Z_{k}[J]=\int
{\mathcal {D}}\phi \exp \left(-S[\phi ]-{\frac
{1}{2}}\phi \cdot R_{k}\cdot \phi +J\cdot
\phi \right)}
where J is the source field. The Legendre
transform of Wk ordinarily gives the effective
action. However, the action that we started
off with is really S[φ]+1/2 φ⋅Rk⋅φ
and so, to get the effective average action,
we subtract off 1/2 φ⋅Rk⋅φ. In other
words,
ϕ
[
J
;
k
]
=
δ
W
k
δ
J
[
J
]
{\displaystyle \phi [J;k]={\frac {\delta W_{k}}{\delta
J}}[J]}
can be inverted to give Jk[φ] and we define
the effective average action Γk as
Γ
k
[
ϕ
]
=
d
e
f
(
−
W
[
J
k
[
ϕ
]
]
+
J
k
[
ϕ
]
⋅
ϕ
)
−
1
2
ϕ
⋅
R
k
⋅
ϕ
.
{\displaystyle \Gamma _{k}[\phi ]\ {\stackrel
{\mathrm {def} }{=}}\ \left(-W\left[J_{k}[\phi
]\right]+J_{k}[\phi ]\cdot \phi \right)-{\tfrac
{1}{2}}\phi \cdot R_{k}\cdot \phi .}
Hence,
d
d
k
Γ
k
[
ϕ
]
=
−
d
d
k
W
k
[
J
k
[
ϕ
]
]
−
δ
W
k
δ
J
⋅
d
d
k
J
k
[
ϕ
]
+
d
d
k
J
k
[
ϕ
]
⋅
ϕ
−
1
2
ϕ
⋅
d
d
k
R
k
⋅
ϕ
=
−
d
d
k
W
k
[
J
k
[
ϕ
]
]
−
1
2
ϕ
⋅
d
d
k
R
k
⋅
ϕ
=
1
2
⟨
ϕ
⋅
d
d
k
R
k
⋅
ϕ
⟩
J
k
[
ϕ
]
;
k
−
1
2
ϕ
⋅
d
d
k
R
k
⋅
ϕ
=
1
2
Tr
⁡
[
(
δ
J
k
δ
ϕ
)
−
1
⋅
d
d
k
R
k
]
=
1
2
Tr
⁡
[
(
δ
2
Γ
k
δ
ϕ
δ
ϕ
+
R
k
)
−
1
⋅
d
d
k
R
k
]
{\displaystyle {\begin{aligned}{\frac {d}{dk}}\Gamma
_{k}[\phi ]&=-{\frac {d}{dk}}W_{k}[J_{k}[\phi
]]-{\frac {\delta W_{k}}{\delta J}}\cdot {\frac
{d}{dk}}J_{k}[\phi ]+{\frac {d}{dk}}J_{k}[\phi
]\cdot \phi -{\tfrac {1}{2}}\phi \cdot {\frac
{d}{dk}}R_{k}\cdot \phi \\&=-{\frac {d}{dk}}W_{k}[J_{k}[\phi
]]-{\tfrac {1}{2}}\phi \cdot {\frac {d}{dk}}R_{k}\cdot
\phi \\&={\tfrac {1}{2}}\left\langle \phi
\cdot {\frac {d}{dk}}R_{k}\cdot \phi \right\rangle
_{J_{k}[\phi ];k}-{\tfrac {1}{2}}\phi \cdot
{\frac {d}{dk}}R_{k}\cdot \phi \\&={\tfrac
{1}{2}}\operatorname {Tr} \left[\left({\frac
{\delta J_{k}}{\delta \phi }}\right)^{-1}\cdot
{\frac {d}{dk}}R_{k}\right]\\&={\tfrac {1}{2}}\operatorname
{Tr} \left[\left({\frac {\delta ^{2}\Gamma
_{k}}{\delta \phi \delta \phi }}+R_{k}\right)^{-1}\cdot
{\frac {d}{dk}}R_{k}\right]\end{aligned}}}
thus
d
d
k
Γ
k
[
ϕ
]
=
1
2
Tr
⁡
[
(
δ
2
Γ
k
δ
ϕ
δ
ϕ
+
R
k
)
−
1
⋅
d
d
k
R
k
]
{\displaystyle {\frac {d}{dk}}\Gamma _{k}[\phi
]={\tfrac {1}{2}}\operatorname {Tr} \left[\left({\frac
{\delta ^{2}\Gamma _{k}}{\delta \phi \delta
\phi }}+R_{k}\right)^{-1}\cdot {\frac {d}{dk}}R_{k}\right]}
is the ERGE which is also known as the Wetterich
equation. As shown by Morris the effective
action Γk is in fact simply related to Polchinski's
effective action Sint via a Legendre transform
relation.
As there are infinitely many choices of Rk,
there are also infinitely many different interpolating
ERGEs.
Generalization to other fields like spinorial
fields is straightforward.
Although the Polchinski ERGE and the effective
average action ERGE look similar, they are
based upon very different philosophies. In
the effective average action ERGE, the bare
action is left unchanged (and the UV cutoff
scale—if there is one—is also left unchanged)
but the IR contributions to the effective
action are suppressed whereas in the Polchinski
ERGE, the QFT is fixed once and for all but
the "bare action" is varied at different energy
scales to reproduce the prespecified model.
Polchinski's version is certainly much closer
to Wilson's idea in spirit. Note that one
uses "bare actions" whereas the other uses
effective (average) actions.
== See also ==
== 
Remarks ==
== Notes
