A gauge theory is a type of theory in physics.
The word gauge means a measurement, a thickness,
an in-between distance, (as in railroad tracks)
or a resulting number of units per certain
parameter (a number of loops in an inch of
fabric or a number of lead balls in a pound
of ammunition). Modern theories describe physical
forces in terms of fields, e.g., the electromagnetic
field, the gravitational field, and fields
that describe forces between the elementary
particles. A general feature of these field
theories is that the fundamental fields cannot
be directly measured; however, some associated
quantities can be measured, such as charges,
energies, and velocities. For example, say
you cannot measure the diameter of a lead
ball, but you can determine how many lead
balls, which are equal in every way, are required
to make a pound. Using the number of balls,
the elemental mass of lead, and the formula
for calculating the volume of a sphere from
its diameter, one could indirectly determine
the diameter of a single lead ball. In field
theories, different configurations of the
unobservable fields can result in identical
observable quantities. A transformation from
one such field configuration to another is
called a gauge transformation; the lack of
change in the measurable quantities, despite
the field being transformed, is a property
called gauge invariance. For example, if you
could measure the color of lead balls and
discover that when you change the color, you
still fit the same number of balls in a pound,
the property of "color" would show gauge invariance.
Since any kind of invariance under a field
transformation is considered a symmetry, gauge
invariance is sometimes called gauge symmetry.
Generally, any theory that has the property
of gauge invariance is considered a gauge
theory.
For example, in electromagnetism the electric
and magnetic fields, E and B are observable,
while the potentials V ("voltage") and A (the
vector potential) are not. Under a gauge transformation
in which a constant is added to V, no observable
change occurs in E or B.
With the advent of quantum mechanics in the
1920s, and with successive advances in quantum
field theory, the importance of gauge transformations
has steadily grown. Gauge theories constrain
the laws of physics, because all the changes
induced by a gauge transformation have to
cancel each other out when written in terms
of observable quantities. Over the course
of the 20th century, physicists gradually
realized that all forces (fundamental interactions)
arise from the constraints imposed by local
gauge symmetries, in which case the transformations
vary from point to point in space and time.
Perturbative quantum field theory (usually
employed for scattering theory) describes
forces in terms of force-mediating particles
called gauge bosons. The nature of these particles
is determined by the nature of the gauge transformations.
The culmination of these efforts is the Standard
Model, a quantum field theory that accurately
predicts all of the fundamental interactions
except gravity.
== History and importance ==
The earliest field theory having a gauge symmetry
was Maxwell's formulation, in 1864–65, of
electrodynamics ("A Dynamical Theory of the
Electromagnetic Field"). The importance of
this symmetry remained unnoticed in the earliest
formulations. Similarly unnoticed, Hilbert
had derived Einstein's equations of general
relativity by postulating a symmetry under
any change of coordinates. Later Hermann Weyl,
inspired by success in Einstein's general
relativity, conjectured (incorrectly, as it
turned out) in year 1919 that invariance under
the change of scale or "gauge" (a term inspired
by the various track gauges of railroads)
might also be a local symmetry of electromagnetism.
Although Weyl's choice of the gauge was incorrect,
the name "gauge" stuck to the approach. After
the development of quantum mechanics, Weyl,
Fock and London modified their gauge choice
by replacing the scale factor with a change
of wave phase, and applying it successfully
to electromagnetism. Gauge symmetry was generalized
mathematically in 1954 by Chen Ning Yang and
Robert Mills in an attempt to describe the
strong nuclear forces. This idea, dubbed Yang–Mills
theory, later found application in the quantum
field theory of the weak force, and its unification
with electromagnetism in the electroweak theory.
The importance of gauge theories for physics
stems from their tremendous success in providing
a unified framework to describe the quantum-mechanical
behavior of electromagnetism, the weak force
and the strong force. This gauge theory, known
as the Standard Model, accurately describes
experimental predictions regarding three of
the four fundamental forces of nature.
== In classical physics ==
=== Electromagnetism ===
Historically, the first example of gauge symmetry
to be discovered was classical electromagnetism.
A static electric field can be described in
terms of an electric potential (voltage) that
is defined at every point in space, and in
practical work it is conventional to take
the Earth as a physical reference that defines
the zero level of the potential, or ground.
But only differences in potential are physically
measurable, which is the reason that a voltmeter
must have two probes, and can only report
the voltage difference between them. Thus
one could choose to define all voltage differences
relative to some other standard, rather than
the Earth, resulting in the addition of a
constant offset. If the potential
V
{\displaystyle V}
is a solution to Maxwell's equations then,
after this gauge transformation, the new potential
V
→
V
+
C
{\displaystyle V\rightarrow V+C}
is also a solution to Maxwell's equations
and no experiment can distinguish between
these two solutions. In other words, the laws
of physics governing electricity and magnetism
(that is, Maxwell equations) are invariant
under gauge transformation. Maxwell's equations
have a gauge symmetry.
Generalizing from static electricity to electromagnetism,
we have a second potential, the magnetic vector
potential A, which can also undergo gauge
transformations. These transformations may
be local. That is, rather than adding a constant
onto V, one can add a function that takes
on different values at different points in
space and time. If A is also changed in certain
corresponding ways, then the same E and B
fields result. The detailed mathematical relationship
between the fields E and B and the potentials
V and A is given in the article Gauge fixing,
along with the precise statement of the nature
of the gauge transformation. The relevant
point here is that the fields remain the same
under the gauge transformation, and therefore
Maxwell's equations are still satisfied.
Gauge symmetry is closely related to charge
conservation. Suppose that there existed some
process by which one could briefly violate
conservation of charge by creating a charge
q at a certain point in space, 1, moving it
to some other point 2, and then destroying
it. We might imagine that this process was
consistent with conservation of energy. We
could posit a rule stating that creating the
charge required an input of energy E1=qV1
and destroying it released E2=qV2, which would
seem natural since qV measures the extra energy
stored in the electric field because of the
existence of a charge at a certain point.
Outside of the interval during which the particle
exists, conservation of energy would be satisfied,
because the net energy released by creation
and destruction of the particle, qV2-qV1,
would be equal to the work done in moving
the particle from 1 to 2, qV2-qV1. But although
this scenario salvages conservation of energy,
it violates gauge symmetry. Gauge symmetry
requires that the laws of physics be invariant
under the transformation
V
→
V
+
C
{\displaystyle V\rightarrow V+C}
, which implies that no experiment should
be able to measure the absolute potential,
without reference to some external standard
such as an electrical ground. But the proposed
rules E1=qV1 and E2=qV2 for the energies of
creation and destruction would allow an experimenter
to determine the absolute potential, simply
by comparing the energy input required to
create the charge q at a particular point
in space in the case where the potential is
V
{\displaystyle V}
and
V
+
C
{\displaystyle V+C}
respectively. The conclusion is that if gauge
symmetry holds, and energy is conserved, then
charge must be conserved.
=== General relativity ===
As discussed above, the gauge transformations
for classical (i.e., non-quantum mechanical)
general relativity are arbitrary coordinate
transformations. Technically, the transformations
must be invertible, and both the transformation
and its inverse must be smooth, in the sense
of being differentiable an arbitrary number
of times.
==== An example of a symmetry in a physical
theory: translation invariance ====
Some global symmetries under changes of coordinate
predate both general relativity and the concept
of a gauge. For example, Galileo and Newton
introduced the notion of translation invariance,
an advancement from the Aristotelian concept
that different places in space, such as the
earth versus the heavens, obeyed different
physical rules.
Suppose, for example, that one observer examines
the properties of a hydrogen atom on Earth,
the other—on the Moon (or any other place
in the universe), the observer will find that
their hydrogen atoms exhibit completely identical
properties. Again, if one observer had examined
a hydrogen atom today and the other—100
years ago (or any other time in the past or
in the future), the two experiments would
again produce completely identical results.
The invariance of the properties of a hydrogen
atom with respect to the time and place where
these properties were investigated is called
translation invariance.
Recalling our two observers from different
ages: the time in their experiments is shifted
by 100 years. If the time when the older observer
did the experiment was t, the time of the
modern experiment is t+100 years. Both observers
discover the same laws of physics. Because
light from hydrogen atoms in distant galaxies
may reach the earth after having traveled
across space for billions of years, in effect
one can do such observations covering periods
of time almost all the way back to the Big
Bang, and they show that the laws of physics
have always been the same.
In other words, if in the theory we change
the time t to t+100 years (or indeed any other
time shift) the theoretical predictions do
not change.
==== Another example of a symmetry: the invariance
of Einstein's field equation under arbitrary
coordinate transformations ====
In Einstein's general relativity, coordinates
like x, y, z, and t are not only "relative"
in the global sense of translations like
t
→
t
+
C
{\displaystyle t\rightarrow t+C}
, rotations, etc., but become completely arbitrary,
so that, for example, one can define an entirely
new time-like coordinate according to some
arbitrary rule such as
t
→
t
+
t
3
/
t
0
2
{\displaystyle t\rightarrow t+t^{3}/t_{0}^{2}}
, where
t
0
{\displaystyle t_{0}}
has units of time, and yet Einstein's equations
will have the same form.Invariance of the
form of an equation under an arbitrary coordinate
transformation is customarily referred to
as general covariance, and equations with
this property are referred to as written in
the covariant form. General covariance is
a special case of gauge invariance.
Maxwell's equations can also be expressed
in a generally covariant form, which is as
invariant under general coordinate transformation
as Einstein's field equation.
== In quantum mechanics ==
=== 
Quantum electrodynamics ===
Until the advent of quantum mechanics, the
only well known example of gauge symmetry
was in electromagnetism, and the general significance
of the concept was not fully understood. For
example, it was not clear whether it was the
fields E and B or the potentials V and A that
were the fundamental quantities; if the former,
then the gauge transformations could be considered
as nothing more than a mathematical trick.
=== Aharonov–Bohm experiment ===
In quantum mechanics, a particle such as an
electron is also described as a wave. For
example, if the double-slit experiment is
performed with electrons, then a wave-like
interference pattern is observed. The electron
has the highest probability of being detected
at locations where the parts of the wave passing
through the two slits are in phase with one
another, resulting in constructive interference.
The frequency of the electron wave is related
to the kinetic energy of an individual electron
particle via the quantum-mechanical relation
E = hf. If there are no electric or magnetic
fields present in this experiment, then the
electron's energy is constant, and, for example,
there will be a high probability of detecting
the electron along the central axis of the
experiment, where by symmetry the two parts
of the wave are in phase.
But now suppose that the electrons in the
experiment are subject to electric or magnetic
fields. For example, if an electric field
was imposed on one side of the axis but not
on the other, the results of the experiment
would be affected. The part of the electron
wave passing through that side oscillates
at a different rate, since its energy has
had −eV added to it, where −e is the charge
of the electron and V the electrical potential.
The results of the experiment will be different,
because phase relationships between the two
parts of the electron wave have changed, and
therefore the locations of constructive and
destructive interference will be shifted to
one side or the other. It is the electric
potential that occurs here, not the electric
field, and this is a manifestation of the
fact that it is the potentials and not the
fields that are of fundamental significance
in quantum mechanics.
==== Explanation with potentials ====
It is even possible to have cases in which
an experiment's results differ when the potentials
are changed, even if no charged particle is
ever exposed to a different field. One such
example is the Aharonov–Bohm effect, shown
in the figure. In this example, turning on
the solenoid only causes a magnetic field
B to exist within the solenoid. But the solenoid
has been positioned so that the electron cannot
possibly pass through its interior. If one
believed that the fields were the fundamental
quantities, then one would expect that the
results of the experiment would be unchanged.
In reality, the results are different, because
turning on the solenoid changed the vector
potential A in the region that the electrons
do pass through. Now that it has been established
that it is the potentials V and A that are
fundamental, and not the fields E and B, we
can see that the gauge transformations, which
change V and A, have real physical significance,
rather than being merely mathematical artifacts.
==== Gauge invariance: the results of the
experiments are independent of the choice
of the gauge for the potentials ====
Note that in these experiments, the only quantity
that affects the result is the difference
in phase between the two parts of the electron
wave. Suppose we imagine the two parts of
the electron wave as tiny clocks, each with
a single hand that sweeps around in a circle,
keeping track of its own phase. Although this
cartoon ignores some technical details, it
retains the physical phenomena that are important
here. If both clocks are sped up by the same
amount, the phase relationship between them
is unchanged, and the results of experiments
are the same. Not only that, but it is not
even necessary to change the speed of each
clock by a fixed amount. We could change the
angle of the hand on each clock by a varying
amount θ, where θ could depend on both the
position in space and on time. This would
have no effect on the result of the experiment,
since the final observation of the location
of the electron occurs at a single place and
time, so that the phase shift in each electron's
"clock" would be the same, and the two effects
would cancel out. This is another example
of a gauge transformation: it is local, and
it does not change the results of experiments.
=== Summary ===
In summary, gauge symmetry attains its full
importance in the context of quantum mechanics.
In the application of quantum mechanics to
electromagnetism, i.e., quantum electrodynamics,
gauge symmetry applies to both electromagnetic
waves and electron waves. These two gauge
symmetries are in fact intimately related.
If a gauge transformation θ is applied to
the electron waves, for example, then one
must also apply a corresponding transformation
to the potentials that describe the electromagnetic
waves. Gauge symmetry is required in order
to make quantum electrodynamics a renormalizable
theory, i.e., one in which the calculated
predictions of all physically measurable quantities
are finite.
=== Types of gauge symmetries ===
The description of the electrons in the subsection
above as little clocks is in effect a statement
of the mathematical rules according to which
the phases of electrons are to be added and
subtracted: they are to be treated as ordinary
numbers, except that in the case where the
result of the calculation falls outside the
range of 0≤θ<360°, we force it to "wrap
around" into the allowed range, which covers
a circle. Another way of putting this is that
a phase angle of, say, 5° is considered to
be completely equivalent to an angle of 365°.
Experiments have verified this testable statement
about the interference patterns formed by
electron waves. Except for the "wrap-around"
property, the algebraic properties of this
mathematical structure are exactly the same
as those of the ordinary real numbers.
In mathematical terminology, electron phases
form an Abelian group under addition, called
the circle group or U(1). "Abelian" means
that addition commutes, so that θ + φ = φ
+ θ. Group means that addition associates
and has an identity element, namely "0". Also,
for every phase there exists an inverse such
that the sum of a phase and its inverse is
0. Other examples of abelian groups are the
integers under addition, 0, and negation,
and the nonzero fractions under product, 1,
and reciprocal.
As a way of visualizing the choice of a gauge,
consider whether it is possible to tell if
a cylinder has been twisted. If the cylinder
has no bumps, marks, or scratches on it, we
cannot tell. We could, however, draw an arbitrary
curve along the cylinder, defined by some
function θ(x), where x measures distance
along the axis of the cylinder. Once this
arbitrary choice (the choice of gauge) has
been made, it becomes possible to detect it
if someone later twists the cylinder.
In 1954, Chen Ning Yang and Robert Mills proposed
to generalize these ideas to noncommutative
groups. A noncommutative gauge group can describe
a field that, unlike the electromagnetic field,
interacts with itself. For example, general
relativity states that gravitational fields
have energy, and special relativity concludes
that energy is equivalent to mass. Hence a
gravitational field induces a further gravitational
field. The nuclear forces also have this self-interacting
property.
=== Gauge bosons ===
Surprisingly, gauge symmetry can give a deeper
explanation for the existence of interactions,
such as the electric and nuclear interactions.
This arises from a type of gauge symmetry
relating to the fact that all particles of
a given type are experimentally indistinguishable
from one another. Imagine that Alice and Betty
are identical twins, labeled at birth by bracelets
reading A and B. Because the girls are identical,
nobody would be able to tell if they had been
switched at birth; the labels A and B are
arbitrary, and can be interchanged. Such a
permanent interchanging of their identities
is like a global gauge symmetry. There is
also a corresponding local gauge symmetry,
which describes the fact that from one moment
to the next, Alice and Betty could swap roles
while nobody was looking, and nobody would
be able to tell. If we observe that Mom's
favorite vase is broken, we can only infer
that the blame belongs to one twin or the
other, but we cannot tell whether the blame
is 100% Alice's and 0% Betty's, or vice versa.
If Alice and Betty are in fact quantum-mechanical
particles rather than people, then they also
have wave properties, including the property
of superposition, which allows waves to be
added, subtracted, and mixed arbitrarily.
It follows that we are not even restricted
to complete swaps of identity. For example,
if we observe that a certain amount of energy
exists in a certain location in space, there
is no experiment that can tell us whether
that energy is 100% A's and 0% B's, 0% A's
and 100% B's, or 20% A's and 80% B's, or some
other mixture. The fact that the symmetry
is local means that we cannot even count on
these proportions to remain fixed as the particles
propagate through space. The details of how
this is represented mathematically depend
on technical issues relating to the spins
of the particles, but for our present purposes
we consider a spinless particle, for which
it turns out that the mixing can be specified
by some arbitrary choice of gauge θ(x), where
an angle θ = 0° represents 100% A and 0%
B, θ = 90° means 0% A and 100% B, and intermediate
angles represent mixtures.
According to the principles of quantum mechanics,
particles do not actually have trajectories
through space. Motion can only be described
in terms of waves, and the momentum p of an
individual particle is related to its wavelength
λ by p = h/λ. In terms of empirical measurements,
the wavelength can only be determined by observing
a change in the wave between one point in
space and another nearby point (mathematically,
by differentiation). A wave with a shorter
wavelength oscillates more rapidly, and therefore
changes more rapidly between nearby points.
Now suppose that we arbitrarily fix a gauge
at one point in space, by saying that the
energy at that location is 20% A's and 80%
B's. We then measure the two waves at some
other, nearby point, in order to determine
their wavelengths. But there are two entirely
different reasons that the waves could have
changed. They could have changed because they
were oscillating with a certain wavelength,
or they could have changed because the gauge
function changed from a 20-80 mixture to,
say, 21-79. If we ignore the second possibility,
the resulting theory doesn't work; strange
discrepancies in momentum will show up, violating
the principle of conservation of momentum.
Something in the theory must be changed.
Again there are technical issues relating
to spin, but in several important cases, including
electrically charged particles and particles
interacting via nuclear forces, the solution
to the problem is to impute physical reality
to the gauge function θ(x). We say that if
the function θ oscillates, it represents
a new type of quantum-mechanical wave, and
this new wave has its own momentum p = h/λ,
which turns out to patch up the discrepancies
that otherwise would have broken conservation
of momentum. In the context of electromagnetism,
the particles A and B would be charged particles
such as electrons, and the quantum mechanical
wave represented by θ would be the electromagnetic
field. (Here we ignore the technical issues
raised by the fact that electrons actually
have spin 1/2, not spin zero. This oversimplification
is the reason that the gauge field θ comes
out to be a scalar, whereas the electromagnetic
field is actually represented by a vector
consisting of V and A.) The result is that
we have an explanation for the presence of
electromagnetic interactions: if we try to
construct a gauge-symmetric theory of identical,
non-interacting particles, the result is not
self-consistent, and can only be repaired
by adding electric and magnetic fields that
cause the particles to interact.
Although the function θ(x) describes a wave,
the laws of quantum mechanics require that
it also have particle properties. In the case
of electromagnetism, the particle corresponding
to electromagnetic waves is the photon. In
general, such particles are called gauge bosons,
where the term "boson" refers to a particle
with integer spin. In the simplest versions
of the theory, gauge bosons are massless,
but it is also possible to construct versions
in which they have mass, as is the case for
the gauge bosons that transmit the nuclear
decay forces
