Global Phase and Physical States

The elementary statement is familiar: multiplying a state vector by one overall phase does not change the physical pure state. The geometric statement is more powerful: a physical pure state is a ray, and choosing a normalized vector on that ray is a gauge choice.

The canonical Core Formalism page is Rays and Global Phase. The chapter guide maps the route from this ray freedom to electromagnetic gauge covariance, holonomy, magnetic translations, and monopole patching. This page uses the ray result to prepare the gauge and geometry viewpoint behind Berry phases, Aharonov–Bohm phases, and phase conventions for families of states.

State Vector Versus Ray

Let $\mathcal H$ be a complex Hilbert space. A nonzero vector $\lvert\psi\rangle$ and any nonzero scalar multiple $\lambda\lvert\psi\rangle$ represent the same pure state ray:

$$\lvert\psi\rangle \sim \lambda\lvert\psi\rangle, \qquad \lambda\in\mathbb C\setminus\{0\}.$$

After normalization, the remaining redundancy is a $U(1)$ phase:

$$\lvert\psi\rangle \sim e^{i\alpha}\lvert\psi\rangle, \qquad \alpha\in\mathbb R.$$

The invariant object can be written as the rank-one projector

$$\Pi_\psi = \lvert\psi\rangle\langle\psi\rvert, \qquad \langle\psi|\psi\rangle=1.$$

Under $\lvert\psi\rangle\mapsto e^{i\alpha}\lvert\psi\rangle$, the projector is unchanged:

$$\Pi_\psi \mapsto e^{i\alpha}\lvert\psi\rangle \langle\psi\rvert e^{-i\alpha} = \Pi_\psi.$$

This is why probabilities, expectation values, and transition probabilities do not depend on the global phase of one isolated state representative.

Phase as a Gauge Choice

The phrase “global phase is unobservable” can be misleading if it suggests that phase conventions are useless. A better statement is:

A single overall phase of a single state vector is not physical, but changes of phase convention over a family of states can carry geometric information.

For one ray, choosing $\lvert\psi\rangle$ rather than $e^{i\alpha}\lvert\psi\rangle$ is like choosing a coordinate label. For a smooth family of rays over a parameter space $M$, choosing normalized vectors $\lvert\psi(R)\rangle$ for each $R\in M$ is a local phase gauge. A different smooth choice has the form

$$\lvert\psi(R)\rangle \mapsto e^{i\chi(R)} \lvert\psi(R)\rangle.$$

The function $\chi(R)$ is not itself observable. But derivatives of the chosen representative appear in Berry-connection formulas. The same issue for wavefunctions over physical space is developed in Local Phase Transformations. With the convention

$$\mathbf A(R) = i\langle\psi(R)|\nabla_R\psi(R)\rangle,$$

the phase-gauge change gives

$$\mathbf A(R) \mapsto \mathbf A(R)-\nabla_R\chi(R).$$

The connection is gauge dependent. Closed-loop holonomies and appropriate curvature integrals are the invariant quantities. This is the same pattern that later appears for electromagnetic gauge potentials: local potentials and phase conventions can change, while properly formed observables do not.

Relative Phase Is Physical

Global phase multiplies all components of a state vector. Relative phase changes interference.

For a two-level state

$$\lvert\psi\rangle = \frac{1}{\sqrt2} \left( \lvert0\rangle + e^{i\phi}\lvert1\rangle \right),$$

the phase $\phi$ is not removable by multiplying the whole state by one number. It affects measurements in bases that compare the two components. In the $\lvert\pm\rangle$ basis,

$$\lvert+\rangle = \frac{\lvert0\rangle+\lvert1\rangle}{\sqrt2}, \qquad P(+) = \frac{1+\cos\phi}{2}.$$

This is the basic interference lesson. A global phase becomes physically relevant only when it is compared with another amplitude, another component, another path, or another branch of an experiment.

Qubit Example

A normalized qubit can be written as

$$\lvert\psi\rangle = e^{i\alpha} \left( \cos\frac{\theta}{2}\lvert0\rangle + e^{i\varphi} \sin\frac{\theta}{2}\lvert1\rangle \right).$$

The angle $\alpha$ is a global phase and does not label a physical point on the Bloch sphere. The angles $\theta$ and $\varphi$ do label the physical ray:

$$\mathbf n = \left( \sin\theta\cos\varphi, \sin\theta\sin\varphi, \cos\theta \right).$$

This is a useful finite-dimensional picture of projective Hilbert space. The state vector lives on a larger sphere of normalized representatives, while the physical ray lives on the space obtained after quotienting by the phase circle.

The same distinction is essential for spinors. A spin-$1/2$ representative changes sign under a $2\pi$ rotation, but the ray is unchanged. The sign matters in interference with another path or rotation history, not as an isolated label of one state.

Time Evolution and Measurable Phases

Under time evolution, a stationary energy eigenstate has

$$\lvert\psi(t)\rangle = e^{-iEt/\hbar} \lvert E\rangle.$$

If this is the entire state and no phase reference is present, the factor $e^{-iEt/\hbar}$ is a global phase. It cannot be read off by measuring that isolated state at one time.

For a superposition,

$$\lvert\psi(t)\rangle = c_1 e^{-iE_1t/\hbar}\lvert E_1\rangle + c_2 e^{-iE_2t/\hbar}\lvert E_2\rangle,$$

the relative phase evolves as

$$\Delta\phi(t) = \Delta\phi(0) - \frac{(E_2-E_1)t}{\hbar}.$$

That relative phase can be measured through interference. This is the practical rule behind many phase-sensitive experiments: absolute phase of one branch is not observed, but phase differences between coherent alternatives are.

Berry phase and the Aharonov–Bohm effect follow the same logic. A cyclic process may return a state to the same ray with a nontrivial phase. That phase is observed when compared with a reference path or another internal component.

From Global Phase to Holonomy

Suppose a normalized state is transported along a closed loop $C$ in a parameter space. The physical ray may return to itself while a chosen representative returns as

$$\lvert\psi(T)\rangle = e^{i\gamma[C]} \lvert\psi(0)\rangle.$$

For a single isolated final state, the phase is again removable. In an interferometer, however, one branch can traverse the loop while another branch provides a reference. The relative phase shift is then observable.

Geometrically, $\gamma[C]$ is a holonomy: the result of comparing phase choices after transport around a loop. The Berry Phase is the canonical adiabatic example. The Aharonov–Bohm Effect is the electromagnetic example where the phase depends on flux enclosed by paths in a multiply connected region.

The important point is not that global phase suddenly becomes directly observable. The point is that a closed-loop phase becomes a relative phase when the experiment supplies a comparison.

Electromagnetic Preview

For a charged wavefunction, a position-dependent phase convention is written schematically as

$$\psi(\mathbf r) \mapsto \exp\left( \frac{iq}{\hbar}\chi(\mathbf r) \right) \psi(\mathbf r).$$

The corresponding electromagnetic potential must transform so that the covariant momentum and Hamiltonian describe the same physics. With the common convention

$$\boldsymbol\pi = \mathbf p-q\mathbf A,$$

the vector potential shifts by a gradient under the matching gauge transformation. The details belong to the later pages in this chapter, but the conceptual seed is already here: phase choices are local redundancies, while gauge-invariant phases around loops can be measured.

Common Mistakes

• Saying “global phase is meaningless” in a way that hides the importance of phase conventions for families of states.
• Treating relative phase as if it were also unobservable.
• Trying to measure the absolute phase of a single isolated state without a reference branch.
• Confusing the gauge-dependent Berry connection with the gauge-invariant Berry phase around a closed loop.
• Forgetting that a phase acquired during time evolution can be physically visible only through comparison.

References

• P. A. M. Dirac, The Principles of Quantum Mechanics, 4th ed., Oxford University Press, 1958.
• A. Peres, Quantum Theory: Concepts and Methods, Kluwer, 1995.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• M. V. Berry, “Quantal phase factors accompanying adiabatic changes,” Proceedings of the Royal Society A 392, 45-57, 1984.
• B. Simon, “Holonomy, the quantum adiabatic theorem, and Berry’s phase,” Physical Review Letters 51, 2167-2170, 1983.
• Y. Aharonov and J. Anandan, “Phase change during a cyclic quantum evolution,” Physical Review Letters 58, 1593-1596, 1987.

Exercises

1. Show that a normalized vector and its phase-rotated representative define the same projector.

Solution

Let $\lvert\psi'\rangle=e^{i\alpha}\lvert\psi\rangle$. Then

$$\lvert\psi'\rangle\langle\psi'\rvert = e^{i\alpha}\lvert\psi\rangle \langle\psi\rvert e^{-i\alpha} = \lvert\psi\rangle\langle\psi\rvert.$$

Therefore the rank-one projector, and hence all probabilities computed from it, are unchanged.

2. For the qubit parametrization above, compute $\langle\sigma_x\rangle$, $\langle\sigma_y\rangle$, and $\langle\sigma_z\rangle$ and verify that $\alpha$ drops out.

Solution

The overall factor $e^{i\alpha}$ cancels between bra and ket. Writing the state without that factor gives

$$\langle\sigma_x\rangle = \sin\theta\cos\varphi, \qquad \langle\sigma_y\rangle = \sin\theta\sin\varphi, \qquad \langle\sigma_z\rangle = \cos\theta.$$

These are the components of the Bloch vector $\mathbf n$ and contain no dependence on the global phase $\alpha$.

3. Derive the gauge transformation of the Berry connection for $\lvert\psi(R)\rangle\mapsto e^{i\chi(R)}\lvert\psi(R)\rangle$ using $\mathbf A=i\langle\psi|\nabla_R\psi\rangle$.

Solution

For $\lvert\psi'\rangle=e^{i\chi}\lvert\psi\rangle$,

$$\nabla_R\lvert\psi'\rangle = e^{i\chi} \left( i(\nabla_R\chi)\lvert\psi\rangle + \nabla_R\lvert\psi\rangle \right).$$

Thus

$$\begin{aligned} \mathbf A' &= i\langle\psi'|\nabla_R\psi'\rangle \\ &= i \left( i\nabla_R\chi + \langle\psi|\nabla_R\psi\rangle \right) \\ &= \mathbf A-\nabla_R\chi. \end{aligned}$$

For a single-valued phase convention on a closed loop, the integral of $\nabla_R\chi$ around the loop vanishes. More generally, the holonomy is defined modulo $2\pi$.