Skip to main content

Mathematics for Physics

243 Conjugate Symmetry

A property of an inner product under which swapping the two vector inputs replaces the scalar value by its complex conjugate that is used to keep the norm of a vector a real number.

definition [d] (Conjugate Symmetry = Hermitian Symmetry) A property of a complex inner product \(\langle \cdot, \cdot \rangle\) on a vector space \(\mathbf{V}\) that satisfies the following condition for all vectors:

  • \(\langle \mathbf{u}, \mathbf{v} \rangle = \overline{\langle \mathbf{v}, \mathbf{u} \rangle}\) .

where

  • \(\mathbf{V}\) is a complex vector space.
  • \(\mathbf{u}, \mathbf{v} \in \mathbf{V}\).
  • \(\overline{\,\cdot\,}\) denotes the complex conjugate.

243.1 Elementary Example

243.1.1 Simple

Conjugate symmetry says swapping the two inputs conjugates the inner-product value.

\[ \langle u,v \rangle = \overline{\langle v,u \rangle} \]

\[ u = (1,0),\quad v = (i,0),\quad \langle u,v \rangle = -i,\quad \langle v,u \rangle = i \]

where

  • \(\langle \cdot,\cdot \rangle\) is the inner product.
  • \(\overline{\,\cdot\,}\) is complex conjugation.

243.1.2 General

On \(\mathbb{C}^{3}\) with the standard Hermitian inner product, the same identity holds for every pair.

\[ \langle u,v \rangle = u_{1}\overline{v_{1}} + u_{2}\overline{v_{2}} + u_{3}\overline{v_{3}} \]

\[ \langle u,v \rangle = \overline{\langle v,u \rangle} \]

where

  • \(u = (u_{1},u_{2},u_{3})\) and \(v = (v_{1},v_{2},v_{3})\) are vectors in \(\mathbb{C}^{3}\).

243.2 References

  1. Griffel, D. H. Applied Functional Analysis. Ellis Horwood, 1981. — inner-product axiom (b) (conjugation under order reversal).
  2. Carroll, S. Spacetime and Geometry: An Introduction to General Relativity. Cambridge University Press, 2021. — Hilbert-space inner product (order reversal = complex conjugation).