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, 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 $U(1)$, but the term often refers to a scalar recording a phase choice or convention, or an ambiguity in choosing a representative.

Properties

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

Phase ambiguity

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

In quantum mechanics, a phase factor is a complex coefficient e^iθ that multiplies a ket $|\psi \rangle$ or bra $\langle \phi |$. 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. That is, the values of $\langle \phi |A|\phi \rangle$ and $\langle \psi |A|\psi \rangle$, where $|\psi \rangle =e^{i\theta }|\phi \rangle$, are the same.

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".

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. For example, this convention is used for the Clebsch–Gordan coefficients.

Phase differences

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

Projective representations and lifts

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 $$U(g)U(h)=\omega (g,h)U(gh),$$ where $\omega (g,h)$ 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.

See also

• Berry phase
• Bra-ket notation
• Euler's formula
• Phasor
• Plane wave
• The circle group U(1)