Gauge transformation
Introduction
A gauge transformation is a mathematical procedure used in physics, particularly in the fields of quantum mechanics and field theory, to change the values of certain quantities without altering the physical predictions of a theory. The term "gauge" originates from the concept of a "gauge invariance" or "gauge symmetry", which is a type of symmetry that the laws of physics exhibit under certain transformations.
Gauge Transformations in Quantum Mechanics
In quantum mechanics, gauge transformations are used to change the phase of the wave function. This is a complex-valued function that provides a complete description of the quantum state of a particle. The phase of the wave function is not physically observable, so changing it does not affect the physical predictions of quantum mechanics.
A gauge transformation in quantum mechanics is given by the equation:
Ψ' = e^iα Ψ
where Ψ is the original wave function, Ψ' is the transformed wave function, α is a real number representing the phase change, and i is the imaginary unit. This transformation leaves the probability density |Ψ|^2 unchanged, which is the observable quantity in quantum mechanics.
Gauge Transformations in Electromagnetism
In the field of electromagnetism, gauge transformations are used to change the potentials without altering the electric and magnetic fields. The potentials are auxiliary quantities introduced to simplify the equations of electromagnetism, and their values are not physically observable.
A gauge transformation in electromagnetism is given by the equations:
A' = A + ∇φ
Φ' = Φ - ∂φ/∂t
where A is the vector potential, Φ is the scalar potential, A' and Φ' are the transformed potentials, ∇φ is the gradient of a scalar function φ, and ∂φ/∂t is the time derivative of φ. This transformation leaves the electric field E = -∇Φ - ∂A/∂t and the magnetic field B = ∇×A unchanged, which are the observable quantities in electromagnetism.
Gauge Transformations in General Relativity
In the theory of general relativity, gauge transformations are used to change the coordinates without altering the curvature of spacetime. The coordinates are auxiliary quantities introduced to describe the geometry of spacetime, and their values are not physically observable.
A gauge transformation in general relativity is given by the equation:
x'^μ = x^μ + ξ^μ
where x^μ are the original coordinates, x'^μ are the transformed coordinates, and ξ^μ is a vector field representing the coordinate change. This transformation leaves the metric tensor g_μν, which describes the curvature of spacetime, unchanged.
Gauge Theories
Gauge transformations are a fundamental concept in gauge theories, which are a class of physical theories based on the principle of gauge invariance. This principle states that the laws of physics should be invariant under gauge transformations, i.e., they should not depend on the choice of the auxiliary quantities like the phase of the wave function, the potentials, or the coordinates.
Examples of gauge theories include quantum electrodynamics (QED), quantum chromodynamics (QCD), and the standard model of particle physics. These theories describe the fundamental forces of nature (except gravity) in terms of gauge fields, which are mathematical objects that transform under gauge transformations.