Math 2.0

Quantum theory Definition

Wick rotation

Builds on Analytic continuationSelf-adjoint operatorSpectral theorem for unbounded self-adjoint operators

Definition

Let d≥1d \ge 1 and let (t,𝐱)∈ℝ×ℝd(t, \mathbf{x}) \in \R \times \R^d be coordinates on Minkowski space with quadratic form t2−∣𝐱∣2t^2 - \abs{\mathbf{x}}^2. The Wick rotation is the substitution t=−iτt = -i\tau with τ∈ℝ\tau \in \R. It sends the Minkowski form to −(τ2+∣𝐱∣2)-(\tau^2 + \abs{\mathbf{x}}^2), the negative of the Euclidean form on ℝd+1\R^{d+1}. Let F(t,𝐱)F(t, \mathbf{x}) be the boundary value on the real tt-axis of a function analytic in tt in the open lower half-plane. Its Wick rotation is the analytic continuation evaluated on the negative imaginary axis t=−iτt = -i\tau, τ>0\tau > 0.

Remarks

Example: let H≥0H \ge 0 be a self-adjoint operator on a Hilbert space. By the spectral theorem, z↦e−izHz \mapsto e^{-izH} is defined and bounded by 11 in norm for Im⁡z≤0\operatorname{Im} z \le 0, strongly continuous there, and analytic in the open lower half-plane. At z=−iτz = -i\tau it gives the contraction semigroup e−τHe^{-\tau H}. So the Wick rotation of the unitary group e−itHe^{-itH} is the semigroup e−τHe^{-\tau H}.

Non-example: let HH be self-adjoint but not bounded below. For Im⁡z<0\operatorname{Im} z < 0, the function ∣e−izE∣=eEIm⁡z\abs{e^{-izE}} = e^{E \operatorname{Im} z} is unbounded on the spectrum, and e−τHe^{-\tau H} is an unbounded operator. So the continuation into the lower half-plane fails as a bounded family.

The sign is a convention: with t=iτt = i\tau one continues into the upper half-plane and uses negative energies. Under the rotation, the Lorentz group of the Minkowski form is traded for the rotation group of Euclidean space.

Reading path

To understand this page from the ground up, read these 21 pages in order. The foundations come first.

Show the path as a graph
Adjoint of an unbounded operator, concept, no evidence levelAdjoint of an unbound…conceptAnalytic continuation, concept, no evidence levelAnalytic continuationconceptBorel sigma-algebra, concept, no evidence levelBorel sigma-algebraconceptContinuous functional calculus, concept, no evidence levelContinuous functional…conceptDensely defined operator, concept, no evidence levelDensely defined opera…conceptHilbert space, concept, no evidence levelHilbert spaceconceptHilbert space adjoint, concept, no evidence levelHilbert space adjointconceptIdentity matrix, object, no evidence levelIdentity matrixobjectMatrix, definition, no evidence levelMatrixdefinitionProjection-valued measure, concept, no evidence levelProjection-valued mea…conceptSelf-adjoint operator, concept, no evidence levelSelf-adjoint operatorconceptSpectrum of an operator, concept, no evidence levelSpectrum of an operat…conceptUnbounded operator, concept, no evidence levelUnbounded operatorconceptTranspose, definition, no evidence levelTransposedefinitionConjugate transpose, definition, no evidence levelConjugate transposedefinitionNormal matrix, definition, no evidence levelNormal matrixdefinitionUnitary matrix, definition, no evidence levelUnitary matrixdefinitionSpectral theorem for normal matrices, statement, no evidence levelSpectral theorem for …statementSpectral theorem for bounded normal operators, statement, no evidence levelSpectral theorem for …statementSpectral theorem for unbounded self-adjoint operators, statement, no evidence levelSpectral theorem for …statementWick rotation, definition, no evidence levelWick rotationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Adjoint of an unbounded operator concept
  2. Analytic continuation concept
  3. Borel sigma-algebra concept
  4. Continuous functional calculus concept
  5. Densely defined operator concept
  6. Hilbert space concept
  7. Hilbert space adjoint concept
  8. Identity matrix object
  9. Matrix definition
  10. Projection-valued measure concept
  11. Self-adjoint operator concept
  12. Spectrum of an operator concept
  13. Unbounded operator concept
  14. Transpose definition
  15. Conjugate transpose definition
  16. Normal matrix definition
  17. Unitary matrix definition
  18. Spectral theorem for normal matrices statement
  19. Spectral theorem for bounded normal operators statement
  20. Spectral theorem for unbounded self-adjoint operators statement
  21. Wick rotation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Wick_rotation
About this page
TagJHXNTZCYQ1
URIm2://JHXNTZCYQ1
Kinddefinition
WritingW0 stub
MSC81T08, 81S40, 81T05
DefinesWick rotation
UsesAnalytic continuation, Self-adjoint operator, Spectral theorem for unbounded self-adjoint operators
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.