Renormalization is a collection of techniques
in quantum field theory, the statistical mechanics
of fields, and the theory of self-similar
geometric structures, that are used to treat
infinities arising in calculated quantities
by altering values of quantities to compensate
for effects of their self-interactions. However,
even if it were the case that no infinities
arise in loop diagrams in quantum field theory,
it can be shown that renormalization of mass
and fields appearing in the original Lagrangian
is necessary.For example, an electron theory
may begin by postulating an electron with
an initial mass and charge. In quantum field
theory a cloud of virtual particles, such
as photons, positrons, and others surrounds
and interacts with the initial electron. Accounting
for the interactions of the surrounding particles
(e.g. collisions at different energies) shows
that the electron-system behaves as if it
had a different mass and charge than initially
postulated. Renormalization, in this example,
mathematically replaces the initially postulated
mass and charge of an electron with the experimentally
observed mass and charge. Mathematics and
experiments prove that positrons and more
massive particles like protons, exhibit precisely
the same observed charge as the electron - even
in the presence of much stronger interactions
and more intense clouds of virtual particles.
Renormalization specifies relationships between
parameters in the theory when parameters describing
large distance scales differ from parameters
describing small distance scales. In high-energy
particle accelerators like the CERN Large
Hadron Collider the concept named pileup occurs
when undesirable proton-proton collisions
interact with data collection for simultaneous,
nearby desirable measurements. Physically,
the pileup of contributions from an infinity
of scales involved in a problem may then result
in further infinities. When describing space-time
as a continuum, certain statistical and quantum
mechanical constructions are not well-defined.
To define them, or make them unambiguous,
a continuum limit must carefully remove "construction
scaffolding" of lattices at various scales.
Renormalization procedures are based on the
requirement that certain physical quantities
(such as the mass and charge of an electron)
equal observed (experimental) values. That
is, the experimental value of the physical
quantity yields practical applications, but
due to their empirical nature the observed
measurement represents areas of quantum field
theory that require deeper derivation from
theoretical bases.
Renormalization was first developed in quantum
electrodynamics (QED) to make sense of infinite
integrals in perturbation theory. Initially
viewed as a suspect provisional procedure
even by some of its originators, renormalization
eventually was embraced as an important and
self-consistent actual mechanism of scale
physics in several fields of physics and mathematics.
Today, the point of view has shifted: on the
basis of the breakthrough renormalization
group insights of Nikolay Bogolyubov and Kenneth
Wilson, the focus is on variation of physical
quantities across contiguous scales, while
distant scales are related to each other through
"effective" descriptions. All scales are linked
in a broadly systematic way, and the actual
physics pertinent to each is extracted with
the suitable specific computational techniques
appropriate for each. Wilson clarified which
variables of a system are crucial and which
are redundant.
Renormalization is distinct from regularization,
another technique to control infinities by
assuming the existence of new unknown physics
at new scales.
== Self-interactions in classical physics
==
The problem of infinities first arose in the
classical electrodynamics of point particles
in the 19th and early 20th century.
The mass of a charged particle should include
the mass-energy in its electrostatic field
(electromagnetic mass). Assume that the particle
is a charged spherical shell of radius re.
The mass–energy in the field is
m
em
=
∫
1
2
E
2
d
V
=
∫
r
e
∞
1
2
(
q
4
π
r
2
)
2
4
π
r
2
d
r
=
q
2
8
π
r
e
,
{\displaystyle m_{\text{em}}=\int {\frac {1}{2}}E^{2}\,dV=\int
_{r_{e}}^{\infty }{\frac {1}{2}}\left({\frac
{q}{4\pi r^{2}}}\right)^{2}4\pi r^{2}\,dr={\frac
{q^{2}}{8\pi r_{e}}},}
which becomes infinite as re → 0. This implies
that the point particle would have infinite
inertia, making it unable to be accelerated.
Incidentally, the value of re that makes
m
em
{\displaystyle m_{\text{em}}}
equal to the electron mass is called the classical
electron radius, which (setting
q
=
e
{\displaystyle q=e}
and restoring factors of c and
ε
0
{\displaystyle \varepsilon _{0}}
) turns out to be
r
e
=
e
2
4
π
ε
0
m
e
c
2
=
α
ℏ
m
e
c
≈
2.8
×
10
−
15
m
,
{\displaystyle r_{e}={\frac {e^{2}}{4\pi \varepsilon
_{0}m_{e}c^{2}}}=\alpha {\frac {\hbar }{m_{e}c}}\approx
2.8\times 10^{-15}~{\text{m}},}
where
α
≈
1
/
137
{\displaystyle \alpha \approx 1/137}
is the fine-structure constant, and
ℏ
/
(
m
e
c
)
{\displaystyle \hbar /(m_{e}c)}
is the Compton wavelength of the electron.
Renormalization: The total effective mass
of a spherical charged particle includes the
actual bare mass of the spherical shell (in
addition to the mass mentioned above associated
with its electric field). If the shell's bare
mass is allowed to be negative, it might be
possible to take a consistent point limit.
This was called renormalization, and Lorentz
and Abraham attempted to develop a classical
theory of the electron this way. This early
work was the inspiration for later attempts
at regularization and renormalization in quantum
field theory.
(See also regularization (physics) for an
alternative way to remove infinities from
this classical problem, assuming new physics
exists at small scales.)
When calculating the electromagnetic interactions
of charged particles, it is tempting to ignore
the back-reaction of a particle's own field
on itself. (Analogous to the back-EMF of circuit
analysis.) But this back-reaction is necessary
to explain the friction on charged particles
when they emit radiation. If the electron
is assumed to be a point, the value of the
back-reaction diverges, for the same reason
that the mass diverges, because the field
is inverse-square.
The Abraham–Lorentz theory had a noncausal
"pre-acceleration." Sometimes an electron
would start moving before the force is applied.
This is a sign that the point limit is inconsistent.
The trouble was worse in classical field theory
than in quantum field theory, because in quantum
field theory a charged particle experiences
Zitterbewegung due to interference with virtual
particle-antiparticle pairs, thus effectively
smearing out the charge over a region comparable
to the Compton wavelength. In quantum electrodynamics
at small coupling, the electromagnetic mass
only diverges as the logarithm of the radius
of the particle.
== Divergences in quantum electrodynamics
==
When developing quantum electrodynamics in
the 1930s, Max Born, Werner Heisenberg, Pascual
Jordan, and Paul Dirac discovered that in
perturbative corrections many integrals were
divergent (see The problem of infinities).
One way of describing the perturbation theory
corrections' divergences was discovered in
1947–49 by Hans Kramers, Hans Bethe,Julian
Schwinger, Richard Feynman, and Shin'ichiro
Tomonaga, and systematized by Freeman Dyson
in 1949. The divergences appear in radiative
corrections involving Feynman diagrams with
closed loops of virtual particles in them.
While virtual particles obey conservation
of energy and momentum, they can have any
energy and momentum, even one that is not
allowed by the relativistic energy–momentum
relation for the observed mass of that particle
(that is,
E
2
−
p
2
{\displaystyle E^{2}-p^{2}}
is not necessarily the squared mass of the
particle in that process, e.g. for a photon
it could be nonzero). Such a particle is called
off-shell. When there is a loop, the momentum
of the particles involved in the loop is not
uniquely determined by the energies and momenta
of incoming and outgoing particles. A variation
in the energy of one particle in the loop
can be balanced by an equal and opposite change
in the energy of another particle in the loop,
without affecting the incoming and outgoing
particles. Thus many variations are possible.
So to find the amplitude for the loop process,
one must integrate over all possible combinations
of energy and momentum that could travel around
the loop.
These integrals are often divergent, that
is, they give infinite answers. The divergences
that are significant are the "ultraviolet"
(UV) ones. An ultraviolet divergence can be
described as one that comes from
the region in the integral where all particles
in the loop have large energies and momenta,
very short wavelengths and high-frequencies
fluctuations of the fields, in the path integral
for the field,
very short proper-time between particle emission
and absorption, if the loop is thought of
as a sum over particle paths.So these divergences
are short-distance, short-time phenomena.
Shown in the pictures at the right margin,
there are exactly three one-loop divergent
loop diagrams in quantum electrodynamics:
(a) A photon creates a virtual electron–positron
pair, which then annihilates. This is a vacuum
polarization diagram.
(b) An electron quickly emits and reabsorbs
a virtual photon, called a self-energy.
(c) An electron emits a photon, emits a second
photon, and reabsorbs the first. This process
is shown in the section below in figure 2,
and it is called a vertex renormalization.
The Feynman diagram for this is also called
a “penguin diagram” due to its shape remotely
resembling a penguin (with the initial and
final state electrons as the arms and legs,
the second photon as the body and the first
looping photon as the head).The three divergences
correspond to the three parameters in the
theory under consideration:
The field normalization Z.
The mass of the electron.
The charge of the electron.The second class
of divergence called an infrared divergence,
is due to massless particles, like the photon.
Every process involving charged particles
emits infinitely many coherent photons of
infinite wavelength, and the amplitude for
emitting any finite number of photons is zero.
For photons, these divergences are well understood.
For example, at the 1-loop order, the vertex
function has both ultraviolet and infrared
divergences. In contrast to the ultraviolet
divergence, the infrared divergence does not
require the renormalization of a parameter
in the theory involved. The infrared divergence
of the vertex diagram is removed by including
a diagram similar to the vertex diagram with
the following important difference: the photon
connecting the two legs of the electron is
cut and replaced by two on-shell (i.e. real)
photons whose wavelengths tend to infinity;
this diagram is equivalent to the bremsstrahlung
process. This additional diagram must be included
because there is no physical way to distinguish
a zero-energy photon flowing through a loop
as in the vertex diagram and zero-energy photons
emitted through bremsstrahlung. From a mathematical
point of view, the IR divergences can be regularized
by assuming fractional differentiation w.r.t.
a parameter, for example:
(
p
2
−
a
2
)
1
2
{\displaystyle \left(p^{2}-a^{2}\right)^{\frac
{1}{2}}}
is well defined at p = a but is UV divergent;
if we take the ​3⁄2-th fractional derivative
with respect to −a2, we obtain the IR divergence
1
p
2
−
a
2
,
{\displaystyle {\frac {1}{p^{2}-a^{2}}},}
so we can cure IR divergences by turning them
into UV divergences.
=== A loop divergence ===
The diagram in Figure 2 shows one of the several
one-loop contributions to electron–electron
scattering in QED. The electron on the left
side of the diagram, represented by the solid
line, starts out with 4-momentum pμ and ends
up with 4-momentum rμ. It emits a virtual
photon carrying rμ − pμ to transfer energy
and momentum to the other electron. But in
this diagram, before that happens, it emits
another virtual photon carrying 4-momentum
qμ, and it reabsorbs this one after emitting
the other virtual photon. Energy and momentum
conservation do not determine the 4-momentum
qμ uniquely, so all possibilities contribute
equally and we must integrate.
This diagram's amplitude ends up with, among
other things, a factor from the loop of
−
i
e
3
∫
d
4
q
(
2
π
)
4
γ
μ
i
(
γ
α
(
r
−
q
)
α
+
m
)
(
r
−
q
)
2
−
m
2
+
i
ϵ
γ
ρ
i
(
γ
β
(
p
−
q
)
β
+
m
)
(
p
−
q
)
2
−
m
2
+
i
ϵ
γ
ν
−
i
g
μ
ν
q
2
+
i
ϵ
.
{\displaystyle -ie^{3}\int {\frac {d^{4}q}{(2\pi
)^{4}}}\gamma ^{\mu }{\frac {i(\gamma ^{\alpha
}(r-q)_{\alpha }+m)}{(r-q)^{2}-m^{2}+i\epsilon
}}\gamma ^{\rho }{\frac {i(\gamma ^{\beta
}(p-q)_{\beta }+m)}{(p-q)^{2}-m^{2}+i\epsilon
}}\gamma ^{\nu }{\frac {-ig_{\mu \nu }}{q^{2}+i\epsilon
}}.}
The various γμ factors in this expression
are gamma matrices as in the covariant formulation
of the Dirac equation; they have to do with
the spin of the electron. The factors of e
are the electric coupling constant, while
the
i
ϵ
{\displaystyle i\epsilon }
provide a heuristic definition of the contour
of integration around the poles in the space
of momenta. The important part for our purposes
is the dependency on qμ of the three big
factors in the integrand, which are from the
propagators of the two electron lines and
the photon line in the loop.
This has a piece with two powers of qμ on
top that dominates at large values of qμ
(Pokorski 1987, p. 122):
e
3
γ
μ
γ
α
γ
ρ
γ
β
γ
μ
∫
d
4
q
(
2
π
)
4
q
α
q
β
(
r
−
q
)
2
(
p
−
q
)
2
q
2
.
{\displaystyle e^{3}\gamma ^{\mu }\gamma ^{\alpha
}\gamma ^{\rho }\gamma ^{\beta }\gamma _{\mu
}\int {\frac {d^{4}q}{(2\pi )^{4}}}{\frac
{q_{\alpha }q_{\beta }}{(r-q)^{2}(p-q)^{2}q^{2}}}.}
This integral is divergent and infinite, unless
we cut it off at finite energy and momentum
in some way.
Similar loop divergences occur in other quantum
field theories.
== Renormalized and bare quantities ==
The solution was to realize that the quantities
initially appearing in the theory's formulae
(such as the formula for the Lagrangian),
representing such things as the electron's
electric charge and mass, as well as the normalizations
of the quantum fields themselves, did not
actually correspond to the physical constants
measured in the laboratory. As written, they
were bare quantities that did not take into
account the contribution of virtual-particle
loop effects to the physical constants themselves.
Among other things, these effects would include
the quantum counterpart of the electromagnetic
back-reaction that so vexed classical theorists
of electromagnetism. In general, these effects
would be just as divergent as the amplitudes
under consideration in the first place; so
finite measured quantities would, in general,
imply divergent bare quantities.
To make contact with reality, then, the formulae
would have to be rewritten in terms of measurable,
renormalized quantities. The charge of the
electron, say, would be defined in terms of
a quantity measured at a specific kinematic
renormalization point or subtraction point
(which will generally have a characteristic
energy, called the renormalization scale or
simply the energy scale). The parts of the
Lagrangian left over, involving the remaining
portions of the bare quantities, could then
be reinterpreted as counterterms, involved
in divergent diagrams exactly canceling out
the troublesome divergences for other diagrams.
=== Renormalization in QED ===
For example, in the Lagrangian of QED
L
=
ψ
¯
B
[
i
γ
μ
(
∂
μ
+
i
e
B
A
B
μ
)
−
m
B
]
ψ
B
−
1
4
F
B
μ
ν
F
B
μ
ν
{\displaystyle {\mathcal {L}}={\bar {\psi
}}_{B}\left[i\gamma _{\mu }\left(\partial
^{\mu }+ie_{B}A_{B}^{\mu }\right)-m_{B}\right]\psi
_{B}-{\frac {1}{4}}F_{B\mu \nu }F_{B}^{\mu
\nu }}
the fields and coupling constant are really
bare quantities, hence the subscript B above.
Conventionally the bare quantities are written
so that the corresponding Lagrangian terms
are multiples of the renormalized ones:
(
ψ
¯
m
ψ
)
B
=
Z
0
ψ
¯
m
ψ
{\displaystyle \left({\bar {\psi }}m\psi \right)_{B}=Z_{0}{\bar
{\psi }}m\psi }
(
ψ
¯
(
∂
μ
+
i
e
A
μ
)
ψ
)
B
=
Z
1
ψ
¯
(
∂
μ
+
i
e
A
μ
)
ψ
{\displaystyle \left({\bar {\psi }}\left(\partial
^{\mu }+ieA^{\mu }\right)\psi \right)_{B}=Z_{1}{\bar
{\psi }}\left(\partial ^{\mu }+ieA^{\mu }\right)\psi
}
(
F
μ
ν
F
μ
ν
)
B
=
Z
3
F
μ
ν
F
μ
ν
.
{\displaystyle \left(F_{\mu \nu }F^{\mu \nu
}\right)_{B}=Z_{3}\,F_{\mu \nu }F^{\mu \nu
}.}
Gauge invariance, via a Ward–Takahashi identity,
turns out to imply that we can renormalize
the two terms of the covariant derivative
piece
ψ
¯
(
∂
+
i
e
A
)
ψ
{\displaystyle {\bar {\psi }}(\partial +ieA)\psi
}
together (Pokorski 1987, p. 115), which is
what happened to Z2; it is the same as Z1.
A term in this Lagrangian, for example, the
electron-photon interaction pictured in Figure
1, can then be written
L
I
=
−
e
ψ
¯
γ
μ
A
μ
ψ
−
(
Z
1
−
1
)
e
ψ
¯
γ
μ
A
μ
ψ
{\displaystyle {\mathcal {L}}_{I}=-e{\bar
{\psi }}\gamma _{\mu }A^{\mu }\psi -(Z_{1}-1)e{\bar
{\psi }}\gamma _{\mu }A^{\mu }\psi }
The physical constant e, the electron's charge,
can then be defined in terms of some specific
experiment: we set the renormalization scale
equal to the energy characteristic of this
experiment, and the first term gives the interaction
we see in the laboratory (up to small, finite
corrections from loop diagrams, providing
such exotica as the high-order corrections
to the magnetic moment). The rest is the counterterm.
If the theory is renormalizable (see below
for more on this), as it is in QED, the divergent
parts of loop diagrams can all be decomposed
into pieces with three or fewer legs, with
an algebraic form that can be canceled out
by the second term (or by the similar counterterms
that come from Z0 and Z3).
The diagram with the Z1 counterterm's interaction
vertex placed as in Figure 3 cancels out the
divergence from the loop in Figure 2.
Historically, the splitting of the "bare terms"
into the original terms and counterterms came
before the renormalization group insight due
to Kenneth Wilson. According to such renormalization
group insights, detailed in the next section,
this splitting is unnatural and actually unphysical,
as all scales of the problem enter in continuous
systematic ways.
=== Running couplings ===
To minimize the contribution of loop diagrams
to a given calculation (and therefore make
it easier to extract results), one chooses
a renormalization point close to the energies
and momenta exchanged in the interaction.
However, the renormalization point is not
itself a physical quantity: the physical predictions
of the theory, calculated to all orders, should
in principle be independent of the choice
of renormalization point, as long as it is
within the domain of application of the theory.
Changes in renormalization scale will simply
affect how much of a result comes from Feynman
diagrams without loops, and how much comes
from the remaining finite parts of loop diagrams.
One can exploit this fact to calculate the
effective variation of physical constants
with changes in scale. This variation is encoded
by beta-functions, and the general theory
of this kind of scale-dependence is known
as the renormalization group.
Colloquially, particle physicists often speak
of certain physical "constants" as varying
with the energy of interaction, though in
fact, it is the renormalization scale that
is the independent quantity. This running
does, however, provide a convenient means
of describing changes in the behavior of a
field theory under changes in the energies
involved in an interaction. For example, since
the coupling in quantum chromodynamics becomes
small at large energy scales, the theory behaves
more like a free theory as the energy exchanged
in an interaction becomes large---a phenomenon
known as asymptotic freedom. Choosing an increasing
energy scale and using the renormalization
group makes this clear from simple Feynman
diagrams; were this not done, the prediction
would be the same, but would arise from complicated
high-order cancellations.
For example,
I
=
∫
0
a
1
z
d
z
−
∫
0
b
1
z
d
z
=
ln
⁡
a
−
ln
⁡
b
−
ln
⁡
0
+
ln
⁡
0
{\displaystyle I=\int _{0}^{a}{\frac {1}{z}}\,dz-\int
_{0}^{b}{\frac {1}{z}}\,dz=\ln a-\ln b-\ln
0+\ln 0}
is ill-defined.
To eliminate the divergence, simply change
lower limit of integral into εa and εb:
I
=
ln
⁡
a
−
ln
⁡
b
−
ln
⁡
ε
a
+
ln
⁡
ε
b
=
ln
⁡
a
b
−
ln
⁡
ε
a
ε
b
{\displaystyle I=\ln a-\ln b-\ln {\varepsilon
_{a}}+\ln {\varepsilon _{b}}=\ln {\tfrac {a}{b}}-\ln
{\tfrac {\varepsilon _{a}}{\varepsilon _{b}}}}
Making sure εb/εa → 1, then I = ln a/b.
== Regularization ==
Since the quantity ∞ − ∞ is ill-defined,
in order to make this notion of canceling
divergences precise, the divergences first
have to be tamed mathematically using the
theory of limits, in a process known as regularization
(Weinberg, 1995).
An essentially arbitrary modification to the
loop integrands, or regulator, can make them
drop off faster at high energies and momenta,
in such a manner that the integrals converge.
A regulator has a characteristic energy scale
known as the cutoff; taking this cutoff to
infinity (or, equivalently, the corresponding
length/time scale to zero) recovers the original
integrals.
With the regulator in place, and a finite
value for the cutoff, divergent terms in the
integrals then turn into finite but cutoff-dependent
terms. After canceling out these terms with
the contributions from cutoff-dependent counterterms,
the cutoff is taken to infinity and finite
physical results recovered. If physics on
scales we can measure is independent of what
happens at the very shortest distance and
time scales, then it should be possible to
get cutoff-independent results for calculations.
Many different types of regulator are used
in quantum field theory calculations, each
with its advantages and disadvantages. One
of the most popular in modern use is dimensional
regularization, invented by Gerardus 't Hooft
and Martinus J. G. Veltman, which tames the
integrals by carrying them into a space with
a fictitious fractional number of dimensions.
Another is Pauli–Villars regularization,
which adds fictitious particles to the theory
with very large masses, such that loop integrands
involving the massive particles cancel out
the existing loops at large momenta.
Yet another regularization scheme is the lattice
regularization, introduced by Kenneth Wilson,
which pretends that hyper-cubical lattice
constructs our space-time with fixed grid
size. This size is a natural cutoff for the
maximal momentum that a particle could possess
when propagating on the lattice. And after
doing a calculation on several lattices with
different grid size, the physical result is
extrapolated to grid size 0, or our natural
universe. This presupposes the existence of
a scaling limit.
A rigorous mathematical approach to renormalization
theory is the so-called causal perturbation
theory, where ultraviolet divergences are
avoided from the start in calculations by
performing well-defined mathematical operations
only within the framework of distribution
theory. The disadvantage of the method is
the fact that the approach is quite technical
and requires a high level of mathematical
knowledge.
=== Zeta function regularization ===
Julian Schwinger discovered a relationship
between zeta function regularization and renormalization,
using the asymptotic relation:
I
(
n
,
Λ
)
=
∫
0
Λ
d
p
p
n
∼
1
+
2
n
+
3
n
+
⋯
+
Λ
n
→
ζ
(
−
n
)
{\displaystyle I(n,\Lambda )=\int _{0}^{\Lambda
}dp\,p^{n}\sim 1+2^{n}+3^{n}+\cdots +\Lambda
^{n}\to \zeta (-n)}
as the regulator Λ → ∞. Based on this,
he considered using the values of ζ(−n)
to get finite results. Although he reached
inconsistent results, an improved formula
studied by Hartle, J. Garcia, and based on
the works by E. Elizalde includes the technique
of the zeta regularization algorithm
I
(
n
,
Λ
)
=
n
2
I
(
n
−
1
,
Λ
)
+
ζ
(
−
n
)
−
∑
r
=
1
∞
B
2
r
(
2
r
)
!
a
n
,
r
(
n
−
2
r
+
1
)
I
(
n
−
2
r
,
Λ
)
,
{\displaystyle I(n,\Lambda )={\frac {n}{2}}I(n-1,\Lambda
)+\zeta (-n)-\sum _{r=1}^{\infty }{\frac {B_{2r}}{(2r)!}}a_{n,r}(n-2r+1)I(n-2r,\Lambda
),}
where the B's are the Bernoulli numbers and
a
n
,
r
=
Γ
(
n
+
1
)
Γ
(
n
−
2
r
+
2
)
.
{\displaystyle a_{n,r}={\frac {\Gamma (n+1)}{\Gamma
(n-2r+2)}}.}
So every I(m, Λ) can be written as a linear
combination of ζ(−1), ζ(−3), ζ(−5),
..., ζ(−m).
Or simply using Abel–Plana formula we have
for every divergent integral:
ζ
(
−
m
,
β
)
−
β
m
2
−
i
∫
0
∞
d
t
(
i
t
+
β
)
m
−
(
−
i
t
+
β
)
m
e
2
π
t
−
1
=
∫
0
∞
d
p
(
p
+
β
)
m
{\displaystyle \zeta (-m,\beta )-{\frac {\beta
^{m}}{2}}-i\int _{0}^{\infty }dt{\frac {(it+\beta
)^{m}-(-it+\beta )^{m}}{e^{2\pi t}-1}}=\int
_{0}^{\infty }dp\,(p+\beta )^{m}}
valid when m > 0, Here the zeta function is
Hurwitz zeta function and Beta is a positive
real number.
The "geometric" analogy is given by, (if we
use rectangle method) to evaluate the integral
so:
∫
0
∞
d
x
(
β
+
x
)
m
≈
∑
n
=
0
∞
h
m
+
1
ζ
(
β
h
−
1
,
−
m
)
{\displaystyle \int _{0}^{\infty }dx\,(\beta
+x)^{m}\approx \sum _{n=0}^{\infty }h^{m+1}\zeta
\left(\beta h^{-1},-m\right)}
Using Hurwitz zeta regularization plus the
rectangle method with step h (not to be confused
with Planck's constant).
The logarithmic divergent integral has the
regularization
∑
n
=
0
∞
1
n
+
a
=
−
ψ
(
a
)
+
l
o
g
(
a
)
{\displaystyle \sum _{n=0}^{\infty }{\frac
{1}{n+a}}=-\psi (a)+log(a)}
since for the Harmonic series
∑
n
=
0
∞
1
a
n
+
1
{\displaystyle \sum _{n=0}^{\infty }{\frac
{1}{an+1}}}
in the limit
a
→
0
{\displaystyle a\to 0}
we must recover the series
∑
n
=
0
∞
1
=
1
/
2
{\displaystyle \sum _{n=0}^{\infty }1=1/2}
For multi-loop integrals that will depend
on several variables
k
1
,
⋯
,
k
n
{\displaystyle k_{1},\cdots ,k_{n}}
we can make a change of variables to polar
coordinates and then replace the integral
over the angles
∫
d
Ω
{\displaystyle \int d\Omega }
by a sum so we have only a divergent integral,
that will depend on the modulus
r
2
=
k
1
2
+
⋯
+
k
n
2
{\displaystyle r^{2}=k_{1}^{2}+\cdots +k_{n}^{2}}
and then we can apply the zeta regularization
algorithm, the main idea for multi-loop integrals
is to replace the factor
F
(
q
1
,
⋯
,
q
n
)
{\displaystyle F(q_{1},\cdots ,q_{n})}
after a change to hyperspherical coordinates
F(r, Ω) so the UV overlapping divergences
are encoded in variable r. In order to regularize
these integrals one needs a regulator, for
the case of multi-loop integrals, these regulator
can be taken as
(
1
+
q
i
q
i
)
−
s
{\displaystyle \left(1+{\sqrt {q}}_{i}q^{i}\right)^{-s}}
so the multi-loop integral will converge for
big enough s using the Zeta regularization
we can analytic continue the variable s to
the physical limit where s = 0 and then regularize
any UV integral, by replacing a divergent
integral by a linear combination of divergent
series, which can be regularized in terms
of the negative values of the Riemann zeta
function ζ(−m).
== Attitudes and interpretation ==
The early formulators of QED and other quantum
field theories were, as a rule, dissatisfied
with this state of affairs. It seemed illegitimate
to do something tantamount to subtracting
infinities from infinities to get finite answers.
Freeman Dyson argued that these infinities
are of a basic nature and cannot be eliminated
by any formal mathematical procedures, such
as the renormalization method.Dirac's criticism
was the most persistent. As late as 1975,
he was saying:
Most physicists are very satisfied with the
situation. They say: 'Quantum electrodynamics
is a good theory and we do not have to worry
about it any more.' I must say that I am very
dissatisfied with the situation because this
so-called 'good theory' does involve neglecting
infinities which appear in its equations,
ignoring them in an arbitrary way. This is
just not sensible mathematics. Sensible mathematics
involves disregarding a quantity when it is
small – not neglecting it just because it
is infinitely great and you do not want it!Another
important critic was Feynman. Despite his
crucial role in the development of quantum
electrodynamics, he wrote the following in
1985:
The shell game that we play is technically
called 'renormalization'. But no matter how
clever the word, it is still what I would
call a dippy process! Having to resort to
such hocus-pocus has prevented us from proving
that the theory of quantum electrodynamics
is mathematically self-consistent. It's surprising
that the theory still hasn't been proved self-consistent
one way or the other by now; I suspect that
renormalization is not mathematically legitimate.While
Dirac's criticism was based on the procedure
of renormalization itself, Feynman's criticism
was very different. Feynman was concerned
that all field theories known in the 1960s
had the property that the interactions become
infinitely strong at short enough distance
scales. This property called a Landau pole,
made it plausible that quantum field theories
were all inconsistent. In 1974, Gross, Politzer
and Wilczek showed that another quantum field
theory, quantum chromodynamics, does not have
a Landau pole. Feynman, along with most others,
accepted that QCD was a fully consistent theory.The
general unease was almost universal in texts
up to the 1970s and 1980s. Beginning in the
1970s, however, inspired by work on the renormalization
group and effective field theory, and despite
the fact that Dirac and various others—all
of whom belonged to the older generation—never
withdrew their criticisms, attitudes began
to change, especially among younger theorists.
Kenneth G. Wilson and others demonstrated
that the renormalization group is useful in
statistical field theory applied to condensed
matter physics, where it provides important
insights into the behavior of phase transitions.
In condensed matter physics, a physical short-distance
regulator exists: matter ceases to be continuous
on the scale of atoms. Short-distance divergences
in condensed matter physics do not present
a philosophical problem since the field theory
is only an effective, smoothed-out representation
of the behavior of matter anyway; there are
no infinities since the cutoff is always finite,
and it makes perfect sense that the bare quantities
are cutoff-dependent.
If QFT holds all the way down past the Planck
length (where it might yield to string theory,
causal set theory or something different),
then there may be no real problem with short-distance
divergences in particle physics either; all
field theories could simply be effective field
theories. In a sense, this approach echoes
the older attitude that the divergences in
QFT speak of human ignorance about the workings
of nature, but also acknowledges that this
ignorance can be quantified and that the resulting
effective theories remain useful.
Be that as it may, Salam's remark in 1972
seems still relevant
Field-theoretic infinities — first encountered
in Lorentz's computation of electron self-mass
— have persisted in classical electrodynamics
for seventy and in quantum electrodynamics
for some thirty-five years. These long years
of frustration have left in the subject a
curious affection for the infinities and a
passionate belief that they are an inevitable
part of nature; so much so that even the suggestion
of a hope that they may, after all, be circumvented
— and finite values for the renormalization
constants computed — is considered irrational.
Compare Russell's postscript to the third
volume of his autobiography The Final Years,
1944–1969 (George Allen and Unwin, Ltd.,
London 1969), p. 221:In the modern world,
if communities are unhappy, it is often because
they have ignorances, habits, beliefs, and
passions, which are dearer to them than happiness
or even life. I find many men in our dangerous
age who seem to be in love with misery and
death, and who grow angry when hopes are suggested
to them. They think hope is irrational and
that, in sitting down to lazy despair, they
are merely facing facts.In QFT, the value
of a physical constant, in general, depends
on the scale that one chooses as the renormalization
point, and it becomes very interesting to
examine the renormalization group running
of physical constants under changes in the
energy scale. The coupling constants in the
Standard Model of particle physics vary in
different ways with increasing energy scale:
the coupling of quantum chromodynamics and
the weak isospin coupling of the electroweak
force tend to decrease, and the weak hypercharge
coupling of the electroweak force tends to
increase. At the colossal energy scale of
1015 GeV (far beyond the reach of our current
particle accelerators), they all become approximately
the same size (Grotz and Klapdor 1990, p.
254), a major motivation for speculations
about grand unified theory. Instead of being
only a worrisome problem, renormalization
has become an important theoretical tool for
studying the behavior of field theories in
different regimes.
If a theory featuring renormalization (e.g.
QED) can only be sensibly interpreted as an
effective field theory, i.e. as an approximation
reflecting human ignorance about the workings
of nature, then the problem remains of discovering
a more accurate theory that does not have
these renormalization problems. As Lewis Ryder
has put it, "In the Quantum Theory, these
[classical] divergences do not disappear;
on the contrary, they appear to get worse.
And despite the comparative success of renormalisation
theory, the feeling remains that there ought
to be a more satisfactory way of doing things."
== Renormalizability ==
From this philosophical reassessment, a new
concept follows naturally: the notion of renormalizability.
Not all theories lend themselves to renormalization
in the manner described above, with a finite
supply of counterterms and all quantities
becoming cutoff-independent at the end of
the calculation. If the Lagrangian contains
combinations of field operators of high enough
dimension in energy units, the counterterms
required to cancel all divergences proliferate
to infinite number, and, at first glance,
the theory would seem to gain an infinite
number of free parameters and therefore lose
all predictive power, becoming scientifically
worthless. Such theories are called nonrenormalizable.
The Standard Model of particle physics contains
only renormalizable operators, but the interactions
of general relativity become nonrenormalizable
operators if one attempts to construct a field
theory of quantum gravity in the most straightforward
manner (treating the metric in the Einstein–Hilbert
Lagrangian as a perturbation about the Minkowski
metric), suggesting that perturbation theory
is useless in application to quantum gravity.
However, in an effective field theory, "renormalizability"
is, strictly speaking, a misnomer. In nonrenormalizable
effective field theory, terms in the Lagrangian
do multiply to infinity, but have coefficients
suppressed by ever-more-extreme inverse powers
of the energy cutoff. If the cutoff is a real,
physical quantity—that is, if the theory
is only an effective description of physics
up to some maximum energy or minimum distance
scale—then these additional terms could
represent real physical interactions. Assuming
that the dimensionless constants in the theory
do not get too large, one can group calculations
by inverse powers of the cutoff, and extract
approximate predictions to finite order in
the cutoff that still have a finite number
of free parameters. It can even be useful
to renormalize these "nonrenormalizable" interactions.
Nonrenormalizable interactions in effective
field theories rapidly become weaker as the
energy scale becomes much smaller than the
cutoff. The classic example is the Fermi theory
of the weak nuclear force, a nonrenormalizable
effective theory whose cutoff is comparable
to the mass of the W particle. This fact may
also provide a possible explanation for why
almost all of the particle interactions we
see are describable by renormalizable theories.
It may be that any others that may exist at
the GUT or Planck scale simply become too
weak to detect in the realm we can observe,
with one exception: gravity, whose exceedingly
weak interaction is magnified by the presence
of the enormous masses of stars and planets.
== Renormalization schemes ==
In actual calculations, the counterterms introduced
to cancel the divergences in Feynman diagram
calculations beyond tree level must be fixed
using a set of renormalisation conditions.
The common renormalization schemes in use
include:
Minimal subtraction (MS) scheme and the related
modified minimal subtraction (MS-bar) scheme
On-shell scheme
== Application in statistical physics ==
A deeper understanding of the physical meaning
and generalization of the
renormalization process, which goes beyond
the dilatation group of conventional renormalizable
theories, 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 1974, 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.
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.
The most important
information in the RG flow is 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 free field theory,
the theory is said to exhibit quantum triviality.
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.
== See also ==
History of quantum field theory
Quantum triviality
Zeno's paradoxes
