From Euclidean Time to Euclidean QFT

Imaginary time performs two closely related jobs in quantum mechanics. Evolution by $e^{-H\tau/\hbar}$ suppresses high-energy components, while a trace of $e^{-\beta H}$ turns the imaginary-time interval into a closed path of circumference $\beta\hbar$. In quantum field theory, the same operator statements remain true, but each time slice now carries an entire field configuration.

That replacement produces the basic Euclidean-QFT dictionary:

$$\begin{aligned} q(\tau) &\longrightarrow \phi(\tau,\mathbf x), \\ \int\mathcal Dq\, e^{-S_E[q]/\hbar} &\longrightarrow \int\mathcal D\phi\, e^{-S_E[\phi]/\hbar}, \\ \tau\in[0,\beta\hbar] &\longrightarrow S^1_{\beta\hbar}\times\Sigma . \end{aligned}$$

The imaginary-time kernel, Wick rotation, and ground-state projection belong to Euclidean and Imaginary-Time Path Integrals. The closed thermal trace, cyclic coordinate measure, permutation sectors, ring-polymer map, and periodic oscillator determinant belong to Path Integrals for Statistical Mechanics. This page takes those results as input and explains what changes when the dynamical variable is a field, when particle statistics determine temporal boundary conditions, and when Euclidean correlation functions must be related back to Lorentzian observables.

Throughout, $\beta=1/(k_B T)$ and $c=1$. Factors of $\hbar$ are retained where they clarify the length $\beta\hbar$ of the thermal circle.

From Lorentzian to Euclidean Fields

Consider a real scalar field with Lorentzian action

$$S_M[\phi] = \int dt\,d^d x\, \left[ \frac12(\partial_t\phi)^2 - \frac12(\boldsymbol\nabla\phi)^2 - V(\phi) \right].$$

Under a justified contour rotation $t=-i\tau$, the exponent transforms schematically as

$$\frac{i}{\hbar}S_M \quad\longrightarrow\quad -\frac{1}{\hbar}S_E,$$

with Euclidean action

$$S_E[\phi] = \int d\tau\,d^d x\, \left[ \frac12(\partial_\tau\phi)^2 + \frac12(\boldsymbol\nabla\phi)^2 + V(\phi) \right].$$

For

$$V(\phi) = \frac12m^2\phi^2 + \frac{\lambda}{4!}\phi^4,$$

the Euclidean kinetic and potential terms are nonnegative when $m^2\ge0$ and $\lambda\ge0$. This makes the formal weight damping rather than oscillatory:

$$e^{iS_M/\hbar} \quad\longrightarrow\quad e^{-S_E/\hbar}.$$

The word “formal” matters. A Wick rotation is a contour deformation in complex time or energy, not a universal symbol substitution. Singularities, boundary conditions, the state being prepared, and the large-field behavior of the action determine whether the deformation is legitimate. The construction of the field measure and its regulator belongs to From Path Integrals in QM to Field Path Integrals.

Vacuum Projection Becomes a Statement About Wavefunctionals

Let $\lvert\varphi\rangle$ denote a generalized field eigenstate on a spatial slice:

$$\widehat\phi(\mathbf x) \lvert\varphi\rangle = \varphi(\mathbf x) \lvert\varphi\rangle.$$

The Euclidean field kernel is

$$\begin{aligned} K_E[\varphi_f,\mathcal T;\varphi_i,0] &:= \langle\varphi_f\rvert e^{-H\mathcal T/\hbar} \lvert\varphi_i\rangle \\ &= \int_{\substack{ \phi(0,\mathbf x)=\varphi_i(\mathbf x)\\ \phi(\mathcal T,\mathbf x)=\varphi_f(\mathbf x) }} \mathcal D\phi\, e^{-S_E[\phi]/\hbar}. \end{aligned}$$

In a regulated finite-volume theory with energy eigenstates $\lvert n\rangle$, its spectral expansion is

$$K_E = \sum_n \Psi_n[\varphi_f]\, \Psi_n^*[\varphi_i]\, e^{-E_n\mathcal T/\hbar},$$

where

$$\Psi_n[\varphi] := \langle\varphi\vert n\rangle$$

is a Schrödinger-picture wavefunctional. If the vacuum is nondegenerate and the boundary data overlap it, then

$$e^{E_0\mathcal T/\hbar} K_E \longrightarrow \Psi_0[\varphi_f]\Psi_0^*[\varphi_i] \qquad (\mathcal T\to\infty).$$

Thus a long Euclidean cylinder prepares the vacuum just as long imaginary-time evolution projects onto the ground state in ordinary quantum mechanics. A half-space path integral can prepare a vacuum wavefunctional on its boundary. Degenerate vacua, gapless spectra, infinite volume, and topological sectors require more care: the limit may project onto a subspace, depend on boundary conditions, or converge without a simple isolated exponential.

The Thermal Trace Becomes a Thermal Circle

For a scalar field, the canonical partition function is

$$Z(\beta) = \operatorname{Tr}e^{-\beta H}.$$

Taking the trace identifies the initial and final field configurations:

$$\begin{aligned} Z(\beta) &= \int\mathcal D\varphi\, \langle\varphi\rvert e^{-\beta H} \lvert\varphi\rangle \\ &= \int_{\phi(\beta\hbar,\mathbf x) = \phi(0,\mathbf x)} \mathcal D\phi\, e^{-S_E[\phi]/\hbar}. \end{aligned}$$

Euclidean time is therefore compact:

$$\tau \sim \tau+\beta\hbar.$$

If space is $\Sigma$, finite-temperature QFT lives on the Euclidean geometry

$$S^1_{\beta\hbar}\times\Sigma.$$

This statement cleanly separates three common constructions:

Euclidean construction	Geometry or boundary data	Quantity prepared
Fixed-boundary kernel	finite interval with $\varphi_i,\varphi_f$	transition kernel in imaginary time
Vacuum projection	long cylinder or half-space	ground-state wavefunctional or vacuum correlator
Thermal trace	compact circle of length $\beta\hbar$	partition function and thermal correlators

The zero-temperature limit sends $\beta\to\infty$, so the thermal circle decompactifies. Vacuum Euclidean QFT and thermal Euclidean QFT are therefore related, but they are not identical formulations at finite $\beta$.

Statistics Fixes the Temporal Boundary Condition

For bosonic fields, the trace produces periodic boundary conditions:

$$\phi(\tau+\beta\hbar,\mathbf x) = \phi(\tau,\mathbf x).$$

For fermionic fields represented by Grassmann coherent states, the trace produces antiperiodic boundary conditions:

$$\begin{aligned} \psi(\tau+\beta\hbar,\mathbf x) &= -\psi(\tau,\mathbf x), \\ \overline\psi(\tau+\beta\hbar,\mathbf x) &= -\overline\psi(\tau,\mathbf x). \end{aligned}$$

The minus sign is not an arbitrary convention and does not mean that a physical fermion state literally changes sign after elapsed imaginary time. It arises from the graded coherent-state trace and is what gives thermal fermion propagators their Fermi–Dirac structure. Coherent-State Path Integrals Preview derives the finite-mode trace formula and its determinant check. Bosonic and fermionic composite operators inherit boundary behavior from their total fermion parity.

Gauge fields are bosonic and are ordinarily periodic around the thermal circle, but gauge transformations, holonomy around the circle, and gauge fixing add structure not present for a scalar field. Those issues belong to finite-temperature gauge theory rather than to this bridge.

Matsubara Modes

Compact Euclidean time makes temporal frequency discrete. A periodic bosonic field has Fourier expansion

$$\phi(\tau,\mathbf x) = \sum_{n\in\mathbb Z} e^{-i\omega_n\tau} \phi_n(\mathbf x),$$

with bosonic Matsubara frequencies

$$\omega_n = \frac{2\pi n}{\beta\hbar}.$$

An antiperiodic fermionic field has

$$\psi(\tau,\mathbf x) = \sum_{n\in\mathbb Z} e^{-i\nu_n\tau} \psi_n(\mathbf x),$$

where

$$\nu_n = \frac{(2n+1)\pi}{\beta\hbar}.$$

Two consequences are immediate:

• bosons possess a zero-frequency mode $n=0$;
• fermions have no zero-frequency mode.

As $\beta\to\infty$, the frequency spacing vanishes and a bosonic sum approaches a continuous integral:

$$\frac{1}{\beta\hbar} \sum_n f(\omega_n) \longrightarrow \int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\, f(\omega).$$

At high temperature or long spatial distance, the bosonic zero mode can dominate static observables. This is the seed of dimensional reduction: under suitable scale-separation assumptions, a thermal field theory in $d+1$ dimensions admits a $d$-dimensional effective description for its longest-distance modes. Interactions, gauge constraints, infrared divergences, and matching coefficients decide whether that description is quantitatively reliable.

Euclidean Correlation Functions

For an operator $O$, define imaginary-time evolution by

$$O(\tau) = e^{\tau H/\hbar} O e^{-\tau H/\hbar}.$$

A thermal Euclidean two-point function is

$$G_E(\tau) = \frac{1}{Z(\beta)} \operatorname{Tr} \left[ e^{-\beta H} \mathcal T_\tau O(\tau)O(0) \right].$$

Cyclicity of the trace gives the Kubo–Martin–Schwinger relation. For bosonic operators, the imaginary-time-ordered correlator is periodic in $\beta\hbar$; a fermionic two-point function is antiperiodic. These are statements about the ordered thermal correlator, not merely about solving a differential equation on a circle.

At zero temperature, Euclidean vacuum correlation functions are often called Schwinger functions:

$$S_n(x_{E1},\ldots,x_{En}) = \langle0\rvert \mathcal T_\tau \left\{ \phi(x_{E1})\cdots\phi(x_{En}) \right\} \lvert0\rangle.$$

Formally, they have the normalized path-integral representation

$$S_n = \frac{ \displaystyle \int\mathcal D\phi\, \phi(x_{E1})\cdots\phi(x_{En}) e^{-S_E[\phi]/\hbar} }{ \displaystyle \int\mathcal D\phi\, e^{-S_E[\phi]/\hbar} }.$$

The denominator removes vacuum normalization factors. The regulator, boundary conditions, and renormalization prescription remain part of the definition.

From Correlation Functions to QFT Observables explains how Euclidean decay, spectral reconstruction, and continuum extrapolation convert these functions into physical information.

Worked Example: Free Scalar Thermal Propagator

For this example set $\hbar=c=k_B=1$. A free scalar field on the thermal circle has Euclidean action

$$S_E = \frac12 \int_0^\beta d\tau \int d^d x\, \left[ (\partial_\tau\phi)^2 + (\boldsymbol\nabla\phi)^2 + m^2\phi^2 \right].$$

Let

$$E_{\mathbf k} := \sqrt{\mathbf k^2+m^2}.$$

The momentum-space Euclidean propagator is

$$G_E(i\omega_n,\mathbf k) = \frac{1} {\omega_n^2+E_{\mathbf k}^2}, \qquad \omega_n = \frac{2\pi n}{\beta}.$$

For $0\le\tau\le\beta$, summing the Matsubara modes gives

$$\begin{aligned} G_E(\tau,\mathbf k) &= \frac{1}{2E_{\mathbf k}} \big[ \big(1+n_B(E_{\mathbf k})\big) e^{-E_{\mathbf k}\tau} \\ &\qquad\qquad + n_B(E_{\mathbf k}) e^{E_{\mathbf k}\tau} \big], \end{aligned}$$

where

$$n_B(E) := \frac{1}{e^{\beta E}-1}.$$

The two terms describe propagation around the thermal circle in the two possible orientations. The expression agrees at $\tau=0$ and $\tau=\beta$, as periodicity requires. In the zero-temperature limit, $n_B(E)\to0$ and

$$G_E(\tau,\mathbf k) \longrightarrow \frac{e^{-E_{\mathbf k}\tau}} {2E_{\mathbf k}} \qquad (\tau\ge0).$$

Exponential Euclidean decay therefore reveals the excitation energy. In an interacting theory, long-distance decay similarly carries spectral information, although multiparticle thresholds, anomalous dimensions, finite volume, and operator overlap complicate the extraction.

Connection to Statistical Field Theory

The Euclidean weight resembles a Gibbs weight:

$$e^{-S_E[\phi]/\hbar} \quad\leftrightarrow\quad e^{-\beta_{\rm cl}\mathcal H_{\rm cl}[\phi]}.$$

This is more than a visual analogy. Euclidean QFT methods and classical statistical-field-theory methods share partition functions, correlation functions, coarse graining, critical points, and renormalization-group flows. A $d$-dimensional quantum system at finite temperature becomes a Euclidean problem with $d$ spatial dimensions and one compact imaginary-time direction. A zero-temperature relativistic theory has an unbounded Euclidean-time direction and, when continuation is valid, Euclidean rotational symmetry.

The analogy has limits. The formal weight need not define a positive probability measure. Fermion determinants, finite density, topological terms, and some gauge-theory formulations can make the effective weight negative or complex. Even when the regulated measure is positive, taking a continuum limit and reconstructing a unitary Lorentzian theory are additional problems.

Statistical Field Theory Preview develops the classical field-measure side of this correspondence, including coarse-field definitions, sources, fluctuations, dimensional reduction, and the distinction between static measures and dynamics.

Returning to Lorentzian QFT

Euclidean correlators encode real-time information through analyticity and spectral representations. In one common convention,

$$G_E(i\omega_n,\mathbf k) = \int_{-\infty}^{\infty} \frac{d\omega'}{2\pi}\, \frac{\rho(\omega',\mathbf k)} {i\omega_n-\omega'},$$

while the retarded correlator is

$$G_R(\omega,\mathbf k) = \int_{-\infty}^{\infty} \frac{d\omega'}{2\pi}\, \frac{\rho(\omega',\mathbf k)} {\omega-\omega'+i0}.$$

This motivates the continuation

$$i\omega_n \longrightarrow \omega+i0,$$

after the Euclidean function has been identified as the boundary value of the appropriate analytic function. Signs and factors of $i$ vary with Green-function conventions, so the spectral definition must accompany the continuation.

Exact analytic continuation is a structural relation. Numerical continuation from finitely many noisy Euclidean data points is an ill-conditioned inverse problem and requires additional information or regularization. Euclidean Monte Carlo data therefore does not automatically provide precise real-time spectra or transport coefficients.

Reflection Positivity and Reconstruction

A collection of Euclidean-invariant functions is not automatically the analytic continuation of a unitary Lorentzian QFT. One central requirement is reflection positivity. Let $\Theta$ reflect Euclidean time, $\tau\mapsto-\tau$, together with complex conjugation. For suitable functionals $F$ supported at positive Euclidean time, reflection positivity requires

$$\langle (\Theta F)F \rangle_E \ge 0.$$

This condition is the Euclidean shadow of a positive Hilbert-space inner product. Together with regularity, symmetry, Euclidean invariance, and clustering assumptions, the Osterwalder–Schrader framework gives conditions under which Euclidean Schwinger functions reconstruct a Lorentzian quantum field theory.

Reflection positivity is not a routine consequence of writing $e^{-S_E}$. It can fail for an arbitrary discretization, an effective nonlocal action, or a complex weight. Gauge-fixed and fermionic theories also require formulations adapted to their constraints and grading. The reconstruction theorem is therefore a boundary on informal Wick-rotation arguments, not merely an optional mathematical refinement.

What This Bridge Owns

This page owns the transition from imaginary-time quantum mechanics to Euclidean field theory:

• field wavefunctionals and vacuum projection;
• the thermal circle;
• periodic and antiperiodic temporal boundary conditions;
• the field-theory realization of bosonic and fermionic Matsubara modes;
• the relation between Euclidean correlators, statistical field theory, and Lorentzian reconstruction.

It does not rederive the quantum-mechanical Euclidean kernel, define the regulated field measure, or develop full thermal QFT. Use Bosonic and Fermionic Matsubara Frequencies for the QM-level grids, units, and index conventions; Matsubara Formalism Preview for the compact-time transform, frequency sums, and convergence checks; Finite-Temperature QFT Bridge for thermal propagators, loop sums, screening, zero-mode matching, and the Euclidean-to-real-time boundary; Euclidean and Imaginary-Time Path Integrals for the canonical QM derivation; From Path Integrals in QM to Field Path Integrals for the field configuration space and measure; and From Propagators in QM to Propagators in QFT for the distinctions among Feynman, retarded, and Euclidean two-point functions.

Common Mistakes

• Treating Wick rotation as the substitution $t=-i\tau$ without checking singularities or boundary data.
• Confusing a fixed-boundary Euclidean kernel, a vacuum path integral, and a thermal trace.
• Assigning periodic boundary conditions to thermal fermion fields.
• Forgetting the factor of $\hbar$ in the thermal circumference $\beta\hbar$ when not using natural units.
• Assuming $e^{-S_E}$ always defines a positive probability measure.
• Reading a Matsubara correlator as a retarded correlator without analytic continuation and a convention check.
• Assuming numerical Euclidean data determines real-time spectral functions uniquely.
• Believing Euclidean regularization removes the need for renormalization or a continuum-limit analysis.

Cross-Links

• Euclidean and Imaginary-Time Path Integrals gives the canonical quantum-mechanical derivations.
• Matsubara Formalism Preview develops the finite-system operator and frequency-sum workflow before the field-theory handoff.
• Finite-Temperature QFT Bridge develops the thermal-field-theory handoff, including screening, static effective theory, and real-time limits.
• Bosonic and Fermionic Matsubara Frequencies fixes the QM-level formulas, units, index symmetries, and zero-mode conventions.
• Euclidean Time and Imaginary-Time Action develops the tunneling boundary-value problem, background-subtracted action, and fixed-energy exponent.
• From Path Integrals in QM to Field Path Integrals develops field configuration space, measures, and regulators.
• From Propagators in QM to Propagators in QFT compares Euclidean, Feynman, retarded, and other propagators.
• From Sources in QM to Generating Functionals in QFT explains how source derivatives generate field correlators.
• Correlation Functions in Path Integrals introduces ordered, Euclidean, connected, and response correlators.
• From Correlation Functions to QFT Observables maps Euclidean and Lorentzian correlators to observable data.
• Green Functions from QM to QFT gives the broader Green-function dictionary.
• Correlation Functions provides a compact many-body reference.
• Path Integrals provides a compact QFT bridge reference.
• Statistical Mechanics Checklist reviews ensembles, partition functions, and thermal notation.

References

• J. Schwinger, “Euclidean Quantum Electrodynamics,” Physical Review 115, 721–731, 1959, doi:10.1103/PhysRev.115.721.
• T. Matsubara, “A New Approach to Quantum-Statistical Mechanics,” Progress of Theoretical Physics 14, 351–378, 1955, doi:10.1143/PTP.14.351.
• P. C. Martin and J. Schwinger, “Theory of Many-Particle Systems. I,” Physical Review 115, 1342–1373, 1959, doi:10.1103/PhysRev.115.1342.
• K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s Functions,” Communications in Mathematical Physics 31, 83–112, 1973, doi:10.1007/BF01645738.
• K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s Functions II,” Communications in Mathematical Physics 42, 281–305, 1975, doi:10.1007/BF01608978.
• M. Le Bellac, Thermal Field Theory, Cambridge University Press, 1996.
• J. I. Kapusta and C. Gale, Finite-Temperature Field Theory: Principles and Applications, 2nd ed., Cambridge University Press, 2006.
• J. Glimm and A. Jaffe, Quantum Physics: A Functional Integral Point of View, 2nd ed., Springer, 1987.

Exercises

1. Vacuum wavefunctional from a long cylinder

Starting from the spectral representation of $K_E[\varphi_f,\mathcal T;\varphi_i,0]$, show that a long Euclidean cylinder isolates the vacuum wavefunctional when the vacuum is nondegenerate and both boundaries have nonzero vacuum overlap. State what changes for a $g$-fold degenerate ground space.

Solution

Factor out the lowest exponential:

$$\begin{aligned} K_E &= e^{-E_0\mathcal T/\hbar} \bigg[ \Psi_0[\varphi_f]\Psi_0^*[\varphi_i] \\ &\qquad + \sum_{n\gt0} \Psi_n[\varphi_f]\Psi_n^*[\varphi_i] e^{-(E_n-E_0)\mathcal T/\hbar} \bigg]. \end{aligned}$$

If $E_n-E_0\gt0$ for the excited states and the vacuum overlaps are nonzero, multiplying by $e^{E_0\mathcal T/\hbar}$ and taking $\mathcal T\to\infty$ leaves

$$\Psi_0[\varphi_f]\Psi_0^*[\varphi_i].$$

For a $g$-fold degenerate ground space, the limit is the ground-space projector kernel

$$\sum_{a=1}^{g} \Psi_{0a}[\varphi_f] \Psi_{0a}^*[\varphi_i].$$

Boundary conditions or small symmetry-breaking sources may select a particular linear combination only after the relevant limits are specified.

2. Derive the Matsubara frequencies

Insert a mode $e^{-i\omega\tau}$ into periodic and antiperiodic boundary conditions on a circle of length $\beta\hbar$. Derive the two allowed frequency sets.

Solution

Periodicity requires

$$e^{-i\omega(\tau+\beta\hbar)} = e^{-i\omega\tau},$$

so

$$e^{-i\omega\beta\hbar}=1.$$

Therefore

$$\omega = \frac{2\pi n}{\beta\hbar}, \qquad n\in\mathbb Z.$$

Antiperiodicity instead requires

$$e^{-i\nu\beta\hbar}=-1,$$

which gives

$$\nu = \frac{(2n+1)\pi}{\beta\hbar}.$$

The periodic set contains $n=0$; the antiperiodic set does not.

3. Zero-temperature free propagator

In natural units, evaluate

$$G_E(\tau,\mathbf k) = \int_{-\infty}^{\infty} \frac{d\omega}{2\pi}\, \frac{e^{i\omega\tau}} {\omega^2+E_{\mathbf k}^2}.$$

Interpret the large-$\lvert\tau\rvert$ behavior.

Solution

For $\tau\gt0$, close the contour in the upper half-plane. The pole at $\omega=iE_{\mathbf k}$ gives

$$G_E(\tau,\mathbf k) = \frac{e^{-E_{\mathbf k}\tau}} {2E_{\mathbf k}}.$$

For $\tau\lt0$, closing in the lower half-plane gives the same result with $\tau$ replaced by $\lvert\tau\rvert$:

$$G_E(\tau,\mathbf k) = \frac{e^{-E_{\mathbf k}\lvert\tau\rvert}} {2E_{\mathbf k}}.$$

The slowest exponential at large Euclidean separation identifies the lowest energy coupled to the chosen field operator at momentum $\mathbf k$. In an interacting theory, the coefficient measures operator overlap and additional exponentials or continua represent higher states.

4. Check thermal periodicity

For the free scalar result

$$G_E(\tau,\mathbf k) = \frac{ \big(1+n_B\big)e^{-E\tau} + n_B e^{E\tau} }{2E}, \qquad 0\le\tau\le\beta,$$

use $n_B=(e^{\beta E}-1)^{-1}$ to verify $G_E(\beta,\mathbf k)=G_E(0,\mathbf k)$. Then take $\beta\to\infty$.

Solution

The Bose factor obeys

$$\big(1+n_B\big)e^{-\beta E} = n_B, \qquad n_B e^{\beta E} = 1+n_B.$$

Hence

$$G_E(\beta,\mathbf k) = \frac{1+2n_B}{2E} = G_E(0,\mathbf k).$$

As $\beta\to\infty$, $n_B\to0$. For fixed $\tau\ge0$,

$$G_E(\tau,\mathbf k) \to \frac{e^{-E\tau}}{2E},$$

the vacuum Euclidean propagator.

5. Zero modes and dimensional reduction

Explain why a bosonic field can have a static Matsubara mode while a thermal fermion cannot. What does this suggest about their long-distance roles at temperatures large compared with the spatial momentum scale?

Solution

Bosonic frequencies are $\omega_n=2\pi n/\beta$, so $\omega_0=0$. Fermionic frequencies are $\nu_n=(2n+1)\pi/\beta$, whose smallest magnitude is $\pi/\beta$. A bosonic zero mode can therefore remain light when all nonzero temporal modes cost energies of order $T$.

At spatial momenta much smaller than $T$, nonzero Matsubara modes can often be integrated out, leaving an effective theory for static bosonic fields in one fewer dimension. Fermions have no static mode and generally contribute through matching coefficients rather than as long-distance thermal degrees of freedom. This argument identifies a possible effective description; it does not by itself prove that perturbative matching or infrared dynamics is under control.

6. Reflection positivity for a free scalar

Let $F$ be linear in a free scalar field and supported at $\tau\gt0$. In spatial momentum space, show that the reflected quadratic form has the structure

$$\langle(\Theta F)F\rangle_E = \int\frac{d^d k}{(2\pi)^d} \frac{1}{2E_{\mathbf k}} \left| \int_0^\infty d\tau\, e^{-E_{\mathbf k}\tau} f(\tau,\mathbf k) \right|^2.$$

Why is this useful?

Solution

The free mixed-representation propagator is

$$G_E(\tau-\tau',\mathbf k) = \frac{ e^{-E_{\mathbf k}\lvert\tau-\tau'\rvert} }{2E_{\mathbf k}}.$$

Reflection sends the first time argument to $-\tau$. For $\tau,\tau'\gt0$,

$$\lvert-\tau-\tau'\rvert = \tau+\tau',$$

so the kernel factorizes:

$$G_E(-\tau-\tau',\mathbf k) = \frac{ e^{-E_{\mathbf k}\tau} e^{-E_{\mathbf k}\tau'} }{2E_{\mathbf k}}.$$

Inserting this into the smeared two-point function gives the stated absolute square, which is nonnegative. The factorization exhibits how Euclidean reflection defines a positive inner product and prepares the Hilbert-space interpretation needed for Lorentzian reconstruction.