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arXiv:2503.12771 · Beijing Journal of Pure and Applied Mathematics

Bras and kets in Euclidean path integrals

Edward Witten

Published Jan 1, 2026• 1 citation
arXivHTMLPDF

Abstract

Here we discuss the relation between bras and kets in Euclidean path integrals.In a theory lacking time-reversal or reflection symmetry, bras and kets differ by an operator 𝒯 that complex conjugates the wavefunction and reverses the orientation of spacetime.In the presence of time-reversal and reflection symmetry, space is unoriented so 𝒯 cannot be defined, but the time-reversal symmetry T is available instead.An important special case of this concerns the relation between the hermitian inner product ⟨ , ⟩ -linear in one variable, antilinear in the other -that is used to define quantum mechanical probabilities, and the symmetric bilinear operator ( , ) that comes most naturally from the Euclidean path integral.They differ by the action of 𝒯 or T on one side. MSC 2020

DOI
10.4310/bpam.260113013520

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Here we discuss the relation between bras and kets in Euclidean path integrals.In a theory lacking time-reversal or reflection symmetry, bras and kets differ by an operator 𝒯 that complex conjugates the wavefunction and reverses the orientation of spacetime.In the presence of time-reversal and reflection symmetry, space is unoriented so 𝒯 cannot be defined, but the time-reversal symmetry T is available instead.An important special case of this concerns the relation between the hermitian inner product ⟨ , ⟩ -linear in one variable, antilinear in the other -that is used to define quantum mechanical probabilities, and the symmetric bilinear operator ( , ) that comes most naturally from the Euclidean path integral.They differ by the action of 𝒯 or T on one side. MSC 2020
In most introductory courses on quantum mechanics one is taught that the Hamiltonian operator must be Hermitian in order that the energy levels be real and that the theory be unitary (probability conserving). To express the Hermiticity of a Hamiltonian, one writes H = H †, where the symbol † denotes the usual Dirac Hermitian conjugation; that is, transpose and complex conjugate. In the past few years it has been recognized that the requirement of Hermiticity, which is often stated as an axiom of quantum mechanics, may be replaced by the less mathematical and more physical requirement of space – time reflection symmetry (𝒫𝒯 symmetry) without losing any of the essential physical features of quantum mechanics. Theories defined by non-Hermitian 𝒫𝒯-symmetric Hamiltonians exhibit strange and unexpected properties at the classical as well as at the quantum level. This paper explains how the requirement of Hermiticity can be evaded and discusses the properties of some non-Hermitian 𝒫𝒯-symmetric quantum theories.
There is a notational discrepancy in how mathematicians and physicists treat Hilbert spaces. In physics, one often works with finite-dimensional Hilbert spaces treated as ℂ n , and hence wants to write the Hermitian inner product as (x, y) = x ∗ y. This definition dictates the use of a Hermitian inner product that is antilinear in the first argument, the convention followed, for example, in Reed & Simon [246]. This is also the choice adopted in Dirac’s bracket notation, whose usage in quantum mechanics is very widespread. Our notation is chosen to extend-as closely as possible-the traditional notation of standard finite-dimensional matrix algebra to arbitrary complex inner product spaces and associated linear operators. In matrix algebra, column vectors and the corresponding matrices with one column are identical objects, row vectors are the linear functionals, and the adjoint is the conjugate transpose. For example, ℍ = ℍ × = ℂ n is the space of column vectors of size n, the dual space ℍ ∗ is the space of row vectors of size n, and the operator product ϕ ∗ ψ of a row vector ϕ ∗ and a column vector ψ is the standard Hermitian inner product of the column vectors ϕ and ψ. We use Greek lower case letters to write vectors, thus emphasizing their intended use as quantum state vectors in quantum mechanics. On the other hand, mathematicians working on reproducing kernel Hilbert spaces use an inner product (x, y) antilinear in the second argument, related to the physicist’s inner product (x, y) by (x, y) =( ). This is the convention followed, for example, in Rudin [252]. Although the two ways of defining the inner product lead to fully equivalent theories, all details look a bit different, a fact that has to be taken into account when reading the literature on the subject. For example, in the description based on the physical tradition, it is preferable to work with the antidual space in place of the dual space used in the mathematical tradition. In functional analysis, linear operators in Hilbert spaces are usually considered each with their own domain. But many computations in quantum mechanics require the consideration of algebras of operators with a common domain. The latter is a Euclidean space, a dense subspace of a Hilbert space. This space and its antidual play in many respects a more basic role in quantum physics than the Hilbert space itself. Therefore, and to avoid possible confusion caused by the different traditions, we give in the present chapter a self-contained introduction to Euclidean spaces and their associated spaces. All proofs are carried out in detail.
Abstract The subject of Chapter 8 is the fundamental principles of quantum theory, the abstract extension of quantum mechanics. Two of the entities explored are kets and operators, with kets being representations of quantum states as well as a source of wave functions. The quantum box and quantum spin kets are specified, as are the quantum numbers that identify them. Operators are introduced and defined in part as the symbolic representations of observable quantities such as position, momentum and quantum spin. Eigenvalues and eigenkets are defined and discussed, with the former identified as the possible outcomes of a measurement. Bras, the counterpart to kets, are introduced as the means of forming probability amplitudes from kets. Products of operators are examined, as is their role underpinning Heisenberg’s Uncertainty Principle. A variety of symbol manipulations are presented. How measurements are believed to collapse linear superpositions to one term of the sum is explored.
Galilean space-time symmetries are discussed in the context of the Liouville space bra-ket formalism and the Wigner-Weyl representations. Also considered is Liouville conjugation (changing operators into their Hermitian conjugates), owing to which any symmetry transformation can be performed by both a unitary and an antiunitary superoperator. Special reference is made to the symmetries of relaxation superoperators for open systems, emphasis being on the practical significance of Wigner-Weyl as symmetry-adapted epresentations. As a side result, symplectic Fourier transforms are shown to be equivalent to inversions.