Phase factor

In physics and representation theory, a phase factor is a multiplier representing the phase of a wave or the phase difference between two quantities. It is formulated as a unit complex number, that is a complex number with absolute value 1. For a complex number written in polar form (such as r e^iθ), the phase factor is the complex exponential (e^iθ),^1: 24 where the variable θ is the phase and i is the imaginary unit. If a quantity like a scalar, vector, or a matrix (representing a wave, state, or operator) is equal to another quantity times a phase factor, then those two quantities are said to be equivalent up to the phase factor, as it leaves the length (or norm) unchanged. As a set, the possible phase factors form the circle group ${\displaystyle U(1)}$, but the term often refers to a scalar recording a phase choice or convention, or an ambiguity in choosing a representative.

For a phase factor ${\displaystyle z}$ the following hold:^1: 24 $${\displaystyle z=e^{i\theta }=\cos \theta +i\sin \theta }$$ $${\displaystyle z^{*}z=1}$$

Multiplying the equation of a plane wave Ae^{i(k·r − ωt)} by a phase factor r e^iθ shifts the phase of the wave by θ: $${\displaystyle e^{i\theta }A\,e^{i({\mathbf {k} \cdot \mathbf {r} -\omega t})}=A\,e^{i({\mathbf {k} \cdot \mathbf {r} -\omega t+\theta })}.}$$ This phase factor is related to the arbitrary selection of the origin of the time axis.^2: 61

In quantum mechanics, a phase factor is a complex coefficient e^iθ that multiplies a ket ${\displaystyle |\psi \rangle }$ or bra ${\displaystyle \langle \phi |}$. It does not, in itself, have any physical meaning, since the introduction of a phase factor does not change the expectation values of a Hermitian operator; this effect is known as phase ambiguity.^1: 108 That is, the values of ${\displaystyle \langle \phi |A|\phi \rangle }$ and ${\displaystyle \langle \psi |A|\psi \rangle }$, where ${\displaystyle |\psi \rangle =e^{i\theta }|\phi \rangle }$, are the same.³

The phase ambiguity may also be described as a flexibility in the definition of quantum state functions. For example, the eigenfunctions of the angular momentum operator are uniquely defined "except for a phase factor".^4: 61

In defining spherical harmonics for use in quantum mechanics, the phase factor may be selected to have a standard value initially selected by Edward Condon and G.H. Shortley.^54: 61 For example this convention is used for the Clebsch–Gordan_coefficients.

Differences in phase factors between two interacting quantum states can sometimes be measurable, such as in the Berry phase,^2: 131 and the Aharonov-Bohm effect.^6: 231 In optics, the phase factor is an important quantity in the treatment of interference.

Phase factors can appear when a mathematical or physical object is determined only up to the choice of a representative. As noted above, in quantum mechanics, pure states are represented by rays in Hilbert space rather than by individual normalized vectors. Thus the physical states are normalized vectors up to a phase factor. Likewise, a symmetry of the ray space may be represented on Hilbert space by a unitary operator, but such an operator is determined only up to multiplication by a phase factor. Consequently, a symmetry group may act by operators satisfying

${\displaystyle U(g)U(h)=\omega (g,h)U(gh),}$

where ${\displaystyle \omega (g,h)}$ is a phase factor. Such an action is a projective representation.

A related ambiguity occurs in the representation theory of the Heisenberg group. Because of the Stone–von Neumann theorem, an automorphism of the underlying position-momentum space gives a unitary operator of the oscillator representation, but only up to a phase factor. The resulting operators therefore define a projective representation of the symplectic group. Passing to the metaplectic group resolves this ambiguity.⁷