Hermitian operator

In plain terms

Quantum mechanics represents each measurable quantity, such as energy or the spin along a chosen axis, by an operator, a matrix that acts on the state of the system. But the entries of that matrix are complex numbers, numbers with an imaginary part, while every laboratory dial reads a real number, so you can ask which matrices can legitimately stand for something measurable.

The answer is a symmetry condition. Take the matrix, flip it across its main diagonal, and replace every entry by its complex conjugate, the mirror-image number with the sign of its imaginary part reversed, and if the result is the original matrix the operator is Hermitian. Every square matrix has special vectors it merely rescales rather than turning in a new direction, and the rescaling factors are called its eigenvalues and the vectors its eigenvectors. For a Hermitian matrix the eigenvalues are forced to be real, and they're precisely the possible readings of the measurement. The eigenvectors are the states that give each reading with certainty, and they form a complete set of mutually perpendicular directions, which in quantum mechanics means perfectly distinguishable states, so measurement becomes a sorting of the state into bins that never blur into one another.

Drop the condition and all of this falls apart, as outcomes could be complex, the states of certainty need no longer be distinguishable, and no consistent sorting exists. The fact that experimental numbers are real, traced back to its source, is this one algebraic symmetry.

In CSD

Everything measurable in CSD's machine-checked corpus is Hermitian, the observables, the Hamiltonians that generate motion, and the density operators that describe states which are only partly known. The spectral theorem, the result that a Hermitian operator can be broken into its eigenvalues and mutually perpendicular eigenvectors, is used constantly. Its most load-bearing single use is in the work on channels, the maps that describe how a state changes when it interacts with something else, where a density operator is first decomposed into its eigenvectors and only then integrated against, and the bridge between two ways of measuring the size of an operator, by trace and by norm, stands on that step.

The finite-dimensional spectral facts the corpus consumes are proved within it rather than assumed, so this part of the measurement machinery isn't taken on trust from outside the formal development.

Mathematically

A Hermitian matrix is one equal to its own conjugate transpose. Its eigenvalues are real, it's unitarily diagonalisable with mutually orthogonal eigenspaces, and hence it admits the functional calculus, applying real functions to operators through their spectra, that the corpus's entropy and channel chains consume throughout.

One workmanlike convention is that the corpus states Hermiticity as a predicate on plain matrices rather than bundling it into a subtype, so statements stay close to the mathematics, proofs stay close to the library, and the finite-dimensional spectral facts consumed are proved and not assumed.

The name

Charles Hermite (1822-1901) was born at Dieuze in Lorraine with a malformed right foot. The Ecole Polytechnique admitted him and then forced him out over the disability, ending the engineering career he had trained for, so mathematics got him instead. He eventually taught at the Polytechnique that had rejected him, at the Ecole Normale and at the Sorbonne, and Poincare was his student.

His 1873 proof that e is transcendental was the first for any naturally occurring number, and his method handed Lindemann the transcendence of pi nine years later, closing the problem of squaring the circle after two millennia. He never sought the second result himself, writing that he dared not attempt it, which is one of history's more consequential displays of mathematical modesty.

Related entries

• Hilbert spaceHilbert space is the stage on which quantum mechanics is usually performed, and CSD's claim is that it's scenery, the shadow cast by the physical object and not the object itself.
• Pauli matricesThree small matrices carry the entire two-level world, and their refusal to commute is the uncertainty principle in its smallest packaging.
• Loewner orderThere's a workable meaning of bigger for operators, and it brings a strange discovery, as taking square roots respects it while taking squares can break it.
• Stone's theoremAsk only that time evolution never jumps, and differentiability, a generator and the Hamiltonian all follow for free.
• Robertson uncertainty relationRobertson's inequality is the precise form of the uncertainty principle, and in CSD the spreads it bounds are the spreads of quantities genuinely distributed over the underlying state space and not merely abstract expectation values.