Observable

Prerequisites: Canonical Quantum Mechanics. Full path (17 pages)

• Intuitive
• Concrete
• Abstract

Intuitive

An observable is a quantity that can be measured: the position of a particle, its energy, its momentum, the component of its spin along some axis. A physical theory has to say which quantities are observables and what values a measurement of them can give.

In classical mechanics this is simple. The state of a system is a point in phase space, a list of all positions and momenta, and every observable is a function that assigns a number to each such point. Measuring it reads off that number, at least in principle without disturbing the system.

In quantum mechanics the state is a vector in a Hilbert space and an observable is an operator acting on it. This has consequences with no classical counterpart. A measurement can only give certain allowed values, which is why energies of atoms come in discrete levels. Before the measurement the outcome is in general not fixed: the theory predicts only probabilities. And some pairs of observables, like position and momentum, cannot both have sharp values in the same state, so measuring one disturbs the other.

In gauge theories a further restriction appears. Part of the mathematical description is redundant, and only quantities that do not change under gauge transformations can be observed. This is the subject of the Abstract section.

Concrete

In quantum mechanics an observable is represented by a self-adjoint (Hermitian) operator $\hat A$ on the Hilbert space of states. Self-adjointness guarantees that its eigenvalues are real and that eigenvectors for different eigenvalues are orthogonal. For a discrete, non-degenerate spectrum

$$\hat A |a_n\rangle = a_n |a_n\rangle , \qquad \hat A = \sum_n a_n |a_n\rangle\langle a_n| .$$

The rules for a measurement on a normalized state $|\psi\rangle$ are:

• The possible outcomes are the eigenvalues $a_n$.
• The probability of outcome $a_n$ is $P(a_n) = |\langle a_n|\psi\rangle|^2$ (Born rule).
• After the outcome $a_n$ the state is $|a_n\rangle$.

For a degenerate eigenvalue $a$ with eigenspace projector $\hat P_a$ the probability is $\langle\psi|\hat P_a|\psi\rangle$ and the state afterwards is $\hat P_a|\psi\rangle$, normalized. The average over many measurements is the expectation value

$$\langle \hat A\rangle = \sum_n a_n P(a_n) = \langle\psi|\hat A|\psi\rangle .$$

Example. For a spin-$\frac12$ particle, $\hat S_z = \frac{\hbar}{2}\sigma_z$ has eigenvalues $\pm\frac{\hbar}{2}$ with eigenvectors $|\uparrow\rangle$, $|\downarrow\rangle$. In the state $|\psi\rangle = (|\uparrow\rangle + |\downarrow\rangle)/\sqrt2$ both outcomes have probability $\frac12$ and $\langle \hat S_z\rangle = 0$. The same state is an eigenstate of $\hat S_x = \frac{\hbar}{2}\sigma_x$ with eigenvalue $+\frac{\hbar}{2}$, so a measurement of $S_x$ gives $+\frac{\hbar}{2}$ with certainty.

Compatible observables. If $[\hat A,\hat B] = 0$ the two operators have a common basis of eigenvectors, and both can have sharp values at once. Otherwise the standard deviations obey the Robertson inequality

$$\sigma_A\,\sigma_B \geq \frac12 \left|\langle [\hat A,\hat B]\rangle\right| ,$$

which for $[\hat x,\hat p] = i\hbar$ gives $\sigma_x\sigma_p \geq \hbar/2$.

Time evolution. In the Heisenberg picture observables carry the time dependence,

$$\frac{d\hat A}{dt} = \frac{i}{\hbar}[\hat H,\hat A] + \frac{\partial \hat A}{\partial t} ,$$

the quantum counterpart of the classical equation $\frac{dF}{dt} = \{F,H\} + \frac{\partial F}{\partial t}$ for a function $F(q,p,t)$ on phase space, see Poisson bracket. An observable that commutes with $\hat H$ and has no explicit time dependence is conserved.

Observables like position and momentum have a continuous spectrum. Then the sums become integrals over generalized eigenvectors, for example $P(x \in [a,b]) = \int_a^b |\psi(x)|^2 dx$, and $\hat x$, $\hat p$ are unbounded operators defined on a dense domain.

• No-Nonsense Quantum Mechanics by Jakob Schwichtenberg, section "Quantum Operators"

Abstract

In the path integral approach to gauge theory, observables are gauge invariant functions on the space $\mathcal A$ of $G$-connections on $E$, where $G$ denotes the structure group and $E$ the fiber bundle. Therefore, an observable $f$ is a function on the space $\mathcal A / \mathcal G$, of connections modulo gauge transformations.

As a result, vacuum expectation values are no longer defined as integrals with a formal Lebesgue measure on $\mathcal A$, but instead with a measure on $\mathcal A/ \mathcal G$. We obtain this measure by pushing forward the Lebesgue measure on $\mathcal A$ by the map $\mathcal A \to \mathcal A/ \mathcal G$ that sends each connection to its gauge equivalence class, and then $A$ denotes a gauge equivalence class of connections in the integral. Since $\mathcal A$ is infinite dimensional, no Lebesgue measure exists on it, so this description is heuristic.

The simplest example of an observable in gauge theory are Wilson loops.

Take note that this procedure of modding out $\mathcal G$ from $\mathcal A$ is what leads to Ghosts. To do this properly requires to make use of the BRST formalism. (Source: Baez, Munian; Gauge Fields, Knots and Gravity, page 342)

Path to this page

1. Calculus
2. Imaginary Numbers
3. Linear Algebra
4. Functional
5. Hilbert Space
6. Legendre Transformation
7. Newton's Second Law
8. Vector Calculus
9. Newtonian Mechanics
10. Variational Calculus
11. Configuration Space
12. Euler-Lagrange Equations
13. Lagrangian Formalism
14. Phase Space
15. Hamiltonian Formalism
16. Schrödinger Equation
17. Canonical Quantum Mechanics