Bras and Kets in Euclidean Path Integrals

TL;DR

This work clarifies how bras and kets and the Hermitian inner product emerge from Euclidean path integrals, distinguishing cases with and without discrete spacetime symmetries. It introduces orientation reversal and antilinear symmetries (T and CRT) as the bridge between the Euclidean bilinear form and the quantum-mechanical inner product, and analyzes the role of spin/pin structures and unorientable manifolds, including fermions. The results include explicit gauge-theory examples and a thorough treatment of how reflection symmetry reshapes the relation between pairings, along with a discussion of Wick rotation's impact on linearity versus antilinearity. The findings have implications for how probabilities are computed from Euclidean path integrals in theories with gravity, unorientable manifolds, and fermions, and clarify when the two inner products coincide or differ due to orientation and discrete symmetries.

Abstract

Quantum mechanics requires a hermitian inner product <~,~> -- linear in one variable, antilinear in the other -- while the inner product (~,~) that comes most naturally from Euclidean path integrals is linear in each variable. Here we discuss the relation between the two inner products. In a theory with no time-reversal or reflection symmetry, they differ by an operator that complex conjugates the wavefunction and reverses the orientation of space; in the presence of reflection and time-reversal symmetry, space is unoriented so such an operator cannot be defined, but the time-reversal symmetry T is available instead and plays the same role.

Figures (5)

• Figure 1: (a) A Euclidean $D$-manifold $X$ with $n$ boundary components, depicted here with $n=4$. The amplitude has bose/fermi symmetry in the external closed universe states. (b) The special case that $n=2$ and $X=Y\times I$, where $Y$ is a $D-1$-manifold and $I$ is an interval of width $\tau$. In quantum field theory without gravity, the bilinear pairing $(~,~)$ is defined by this path integral in the limit $\tau\to 0$. In topological field theory, there is no dependence on $\tau$. In a gravitational theory, one has to project such a path integral onto states that obey the gravitational constraints.
• Figure 2: $X$ is a reflection-symmetric manifold that can be divided in two parts by cutting on its plane of symmetry $Y$. For any fixed values of the fields on $Y$, the path integral on the part of $X$ above the cut is the complex conjugate of the path integral on the part below the cut, because the reflection symmetry of $X$ reverses the orientation of $X$ and complex conjugates the action and the path integral. Hence for any values of the fields on $Y$, the product of the path integrals above and below $Y$ is positive. Upon integrating over fields on $Y$, this implies that the path integral on $X$ is positive. This property, known as reflection positivity, implies that the norm of the state on $Y$ that is prepared by the path integral on the lower portion of $X$ is positive. This is the basic argument proving positivity of Hilbert space norms in the context of path integrals.
• Figure 3: A state on a $D-1$-manifold $Y$ can be determined by asymptotic data at $Z$. A familiar version of this occurs with negative cosmological constant: $X$ can have Euclidean signature and $Z$ is the conformal boundary of $X$. With zero or positive cosmological constant, $X$ could be an expanding or contracting FRW spacetime in Lorentz signature with $Z$ at the far future or past. In all cases, with generic asymptotic data on $Z$, the induced state on $Y$ is not ${\mathcal{T}}$-invariant.
• Figure 4: A path integral on a $D$-manifold $X$ can be "cut" on any embedded $D-1$-manifold $Y$. In a reflection-invariant theory of gravity, if $Y$ is compact and has an orientation-reversing symmetry $\rho$, one can glue the two sides of the cut back together after acting with $\rho$ on one side. A sum over such "twists" will project onto $\rho$-invariant states. Hence only $\rho$-invariant states can propagate. There is no close analog of this for ${\mathcal{T}}$ or for the time-reversal symmetries considered later.
• Figure 5: A path integral on $Y\times I$, where $I$ is an interval. The Hamiltonian $H$ has been inserted on the submanifold $Y\times p$, with $p$ a point in $I$. As $H$ is conserved, it can be freely moved to the top or bottom of the cylinder, implying that $(H\Psi,\chi)=(\Psi,H\chi)$. Thus $H$ is symmetric with respect to $(~,~)$, as well as hermitian with respect to ${\langle}~,~{\rangle}$. In a time-reversal invariant theory with time-reversal operator ${\sf T}$ that satisfies ${\sf T}^2=1$, a similar argument with ${\sf T}$ inserted on $Y\times p$, taking into account that ${\sf T}$ is antilinear, proves that $({\sf T}\Psi,\chi)=\overline{(\Psi,{\sf T}\chi)}$.