In
quantum field theory
In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, a Ward–Takahashi identity is an identity between
correlation functions that follows from the global or gauge
symmetries of the theory, and which remains valid after
renormalization
Renormalization is a collection of techniques in quantum field theory, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of the ...
.
The Ward–Takahashi identity of
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
(QED) was originally used by
John Clive Ward and
Yasushi Takahashi to relate the
wave function renormalization of the
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
to its
vertex renormalization factor, guaranteeing the cancellation of the
ultraviolet divergence to all orders of
perturbation theory
In mathematics and applied mathematics, perturbation theory comprises methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem. A critical feature of the technique is a middle ...
. Later uses include the extension of the proof of
Goldstone's theorem
In physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu within the context of the BCS superc ...
to all orders of perturbation theory.
More generally, a Ward–Takahashi identity is the quantum version of classical current conservation associated to a continuous symmetry by
Noether's theorem
Noether's theorem states that every continuous symmetry of the action of a physical system with conservative forces has a corresponding conservation law. This is the first of two theorems (see Noether's second theorem) published by the mat ...
. Such symmetries in quantum field theory (almost) always give rise to these generalized Ward–Takahashi identities which impose the symmetry on the level of the quantum mechanical amplitudes. This generalized sense should be distinguished when reading literature, such as
Michael Peskin and
Daniel Schroeder's textbook,
from the original Ward–Takahashi identity.
The detailed discussion below concerns QED, an
abelian theory
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...
to which the Ward–Takahashi identity applies. The equivalent identities for
non-abelian theories such as
quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the study of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type of ...
(QCD) are the
Slavnov–Taylor identities.
The Ward operator describes how a scalar term in a Lagrangian transforms under infinitesimal gauge transformations. It is closely related to the
BRST operator and plays a central role in providing a geometric description of the consistent quantization of
gauge theories
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
.
Ward–Takahashi identity
The Ward–Takahashi identity applies to correlation functions in
momentum space, which do not necessarily have all their external momenta
on-shell. Let
:
be a
QED correlation function involving an external
photon
A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
with momentum ''k'' (where
is the
polarization vector of the photon and summation over
is implied), ''n'' initial-state
electron
The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s with momenta
, and ''n'' final-state electrons with momenta
. Also define
to be the simpler
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
that is obtained by removing the photon with momentum ''k'' from our original amplitude. Then the Ward–Takahashi identity reads
:
where
is the
charge of the electron and is negative in sign. Note that if
has its external electrons on-shell, then the amplitudes on the right-hand side of this identity each have one external particle off-shell, and therefore they do not contribute to
S-matrix
In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...
elements.
Ward identity
The Ward identity is a specialization of the Ward–Takahashi identity to
S-matrix
In physics, the ''S''-matrix or scattering matrix is a Matrix (mathematics), matrix that relates the initial state and the final state of a physical system undergoing a scattering, scattering process. It is used in quantum mechanics, scattering ...
elements, which describe physically possible
scattering
In physics, scattering is a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including particles and radiat ...
processes and thus have all their external particles
on-shell. Again let
be the amplitude for some QED process involving an external photon with momentum
, where
is the
polarization vector of the photon. Then the Ward identity reads:
:
Physically, what this identity means is the longitudinal polarization of the photon which arises in the
ξ gauge is unphysical and disappears from the S-matrix.
Examples of its use include constraining the
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ...
structure of the
vacuum polarization
In quantum field theory, and specifically quantum electrodynamics, vacuum polarization describes a process in which a background electromagnetic field produces virtual electron–positron pairs that change the distribution of charges and curr ...
and of the electron
vertex function in QED.
Derivation in the path integral formulation
In the path integral formulation, the Ward–Takahashi identities are a reflection of the invariance of the
functional measure under a
gauge transformation
In the physics of gauge theory, gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant Degrees of freedom (physics and chemistry), degrees of freedom in field (physics), field variab ...
. More precisely, if
represents a gauge transformation by
(and this applies even in the case where the physical symmetry of the system is
global
Global may refer to:
General
*Globe, a spherical model of celestial bodies
*Earth, the third planet from the Sun
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* ''Global'' (Paul van Dyk album), 2003
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or even nonexistent; we are only worried about the ''invariance of the functional measure'' here), then
:
expresses the invariance of the functional measure where
is the
action and
is a
functional of the
fields. If the gauge transformation corresponds to a ''
global
Global may refer to:
General
*Globe, a spherical model of celestial bodies
*Earth, the third planet from the Sun
Entertainment
* ''Global'' (Paul van Dyk album), 2003
* ''Global'' (Bunji Garlin album), 2007
* ''Global'' (Humanoid album), 198 ...
'' symmetry of the theory, then,
:
for some "
current" J (as a functional of the fields
) after
integrating by parts and assuming that the
surface terms can be neglected.
Then, the Ward–Takahashi identities become
:
This is the QFT analog of the
Noether continuity equation .
If the gauge transformation corresponds to an actual
gauge symmetry
In physics, a gauge theory is a type of field theory in which the Lagrangian, and hence the dynamics of the system itself, does not change under local transformations according to certain smooth families of operations (Lie groups). Formally, t ...
then
:
where
is the gauge invariant action and
is a non-gauge-invariant
gauge fixing term. Gauge-fixing terms are required so as to be able to perform
second quantization
Second quantization, also referred to as occupation number representation, is a formalism used to describe and analyze quantum many-body systems. In quantum field theory, it is known as canonical quantization, in which the fields (typically as ...
of a classical gauge theory. The path-integral (Lagrangian) formulation of quantum field theory does not entirely avoid the need for gauge-fixing, as there is still a need to compute the asymptotic states of the
scattering matrix (''e.g'' in the
interaction picture.) In short, gauge-fixing is required, but it breaks the overall gauge invariance of the theory. The Ward–Takahashi identities then describe exactly how all of the different fields are tied to one-another, under an infinitessimal gauge transformation. These Ward–Takahashi identities are generated by the Ward operator; in the linearized form, the Ward operator is the
BRST operator. The corresponding
charge is the
BRST charge. When the gauge theory is formulated on a
fiber bundle
In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a pr ...
, the Ward–Takahashi identities correspond to a (global) right-action in the
principle bundle: they are generated by the
Lie derivative on the
vertical bundle
In mathematics, the vertical bundle and the horizontal bundle are Vector bundle, vector bundles associated to a Fiber bundle#Differentiable fiber bundles, smooth fiber bundle. More precisely, given a smooth fiber bundle \pi\colon E\to B, the verti ...
.
When the functional measure is not gauge invariant, but happens to satisfy
:
with
is some functional of the fields
, the corresponding relation gives the anomalous Ward–Takahashi identity. The conventional example is the
chiral anomaly
In theoretical physics, a chiral anomaly is the anomalous nonconservation of a chiral current. In everyday terms, it is analogous to a sealed box that contained equal numbers of left and right-handed bolts, but when opened was found to have mor ...
. This example is prominent in the
sigma model theory of
nuclear force
The nuclear force (or nucleon–nucleon interaction, residual strong force, or, historically, strong nuclear force) is a force that acts between hadrons, most commonly observed between protons and neutrons of atoms. Neutrons and protons, both ...
s. In this theory, the
neutron
The neutron is a subatomic particle, symbol or , that has no electric charge, and a mass slightly greater than that of a proton. The Discovery of the neutron, neutron was discovered by James Chadwick in 1932, leading to the discovery of nucle ...
and
proton
A proton is a stable subatomic particle, symbol , Hydron (chemistry), H+, or 1H+ with a positive electric charge of +1 ''e'' (elementary charge). Its mass is slightly less than the mass of a neutron and approximately times the mass of an e ...
, in an
isospin
In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle.
Isospin is also known as isobaric spin or isotopic spin.
Isospin symmetry is a subset of the flavour symmetr ...
doublet, feel forces mediated by
pion
In particle physics, a pion (, ) or pi meson, denoted with the Greek alphabet, Greek letter pi (letter), pi (), is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the ...
s, in an isospin triplet. This theory has not one, but two distinct global symmetries: the vector
and the axial vector
symmetries; equivalently, the left and right
chiral symmetries. The corresponding currents are the
isovector current (the
rho meson
In particle physics, a rho meson is a short-lived hadronic particle that is an isospin triplet whose three states are denoted as , and . Along with pions and omega mesons, the rho meson carries the nuclear force within the atomic nucleus. Afte ...
) and the
axial vector current. It is not possible to quantize both at the same time (due to the anomalous Ward–Takahashi identity); by convention, the vector symmetry is quantized so that the vector current is conserved, while the axial vector current is not conserved. The
rho meson
In particle physics, a rho meson is a short-lived hadronic particle that is an isospin triplet whose three states are denoted as , and . Along with pions and omega mesons, the rho meson carries the nuclear force within the atomic nucleus. Afte ...
is then interpreted as the
gauge boson of the vector symmetry, whereas the axial symmetry is
spontaneously broken. The breaking is due to quantization, that is, due to the anomalous Ward–Takahashi identity (rather than to a Higgs-style Mexican-hat potential, which results in an entirely different kind of symmetry breaking). The divergence of the axial current relates the
pion-nucleon interaction to pion decay, fixing
as the
axial coupling constant. The
Goldberger–Treiman relation relates
to the
pion decay constant . In this way, the chiral anomaly provides the canonical description of the pion-nuclean interaction.
References
{{DEFAULTSORT:Ward-Takahashi identity
Gauge theories
Quantum electrodynamics