Finite-Mode Method (FM²): A Rigorous Lorentzian Path Integral Framework and its Implications for Induced Gravity

Finite-Mode Path Integration in Curved Spacetime: From Real-Time Stability to the Emergence of Geometry via Induced Gravity Yu-Wei Chen Department of Physics, National Cheng Kung University, Tainan 701, Taiwan [email protected] November 24, 2025 Abstract The Feynman path integral, while constituting the cornerstone of modern quantum field theory, suffers from severe measure-theoretic pathologies in Minkowski spacetime, particularly in time-dependent cosmological backgrounds where the traditional Wick rotation becomes ill-defined due to the complexification of the metric. In this work, we present the Finite-Mode Method (FM2), a rigorous constructive formalism for real-time path integration. By projecting the quantum field onto a complete Gabor frame—explicitly defined as the product of a Gaussian window and a plane wave—we regularize the infinite-dimensional integral into a sequence of convergent finite-dimensional matrix integrals. We solve the convergence problem of oscillatory Fresnel integrals with indefinite quadratic forms by applying Lefschetz Thimble theory, explicitly demonstrating the contour rotations required for both spacelike and timelike modes. We prove that the spacetime metric is not a fundamental background but emerges as the effective refractive index of the vacuum, rigorously satisfying the Hadamard singularity condition. The formalism is validated by non-perturbatively deriving the exact Mehler kernel for the quantum harmonic oscillator via the Gel’fand-Yaglom theorem and the (1+1)D Casimir energy via Euler-Maclaurin expansion. Finally, utilizing the Heat Kernel expansion of the FM2effective action, we provide a first-principles derivation of the Einstein Field Equations, demonstrating that gravity emerges as a statistical entropic force induced by the Gabor-truncated quantum fluctuations of matter fields. Contents 1 Introduction: The Crisis of Wick Rotation 3 2 Methodology: The Finite-Mode Method (FM2)3 2.1 Failure of the Pure Fourier Basis ........................... 3 2.2 The Gabor Frame and Field Expansion ....................... 4 2.3 From the Action to a Finite-Dimensional Quadratic Form ............. 5 2.4 Lefschetz Thimble Regularization of the Fresnel Integral .............. 6 2.5 Reconstruction of the Full Path Integral ....................... 7 3 Optical Analogy: Effective Metric as Refractive Index 7 1 4 Verification I: The Quantum Harmonic Oscillator 7 4.1 The Gel’fand–Yaglom Theorem ............................ 8 4.2 FM2Determinant Limit and Mehler Kernel ..................... 8 5 Verification II: (1+1)D Casimir Effect from Matrix Trace 9 5.1 FM2Matrix Construction and Energy Definition .................. 9 5.2 Euler–Maclaurin Expansion .............................. 9 6 Emergence of Gravity: Induced Einstein Field Equations 10 6.1 Step 1: Matter Action without Bare Gravity .................... 10 6.2 Step 2: Schwinger Proper Time and Heat Kernel .................. 10 6.3 Step 3: Variational Principle and Einstein Equations ................ 11 7 Conclusion 12 2 1 Introduction: The Crisis of Wick Rotation The Feynman path integral formally defines the transition amplitude as a sum over all possible field configurations: Z=ZDϕ e i ℏS[ϕ].(1) In standard Quantum Field Theory (QFT), mathematical rigor is typically sought via Wick rotation to Euclidean time (t→ −iτ). This relies on the assumption that the Euclidean action SEis positive definite, allowing the oscillatory factor eiS to become a Boltzmann weight e−SE. However, in Quantum Cosmology, this procedure is fundamentally flawed. Consider a Friedmann-Robertson-Walker (FRW) universe with a scale factor a(t) = 1 + ϵt. Under Wick rotation, the metric determinant becomes complex: √−g∝a(t)3t→−iτ −−−−→ (1 −iϵτ)3.(2) The real part of the Euclidean measure factor, Re[(1 −iϵτ)3]=1−3ϵ2τ2, becomes negative for Euclidean times τ > 1/(√3ϵ). Consequently, the ”Euclidean” kinetic term flips sign, rendering the action unbounded from below and causing the path integral to diverge catastrophically. To resolve this, a strictly real-time formalism is required. 2 Methodology: The Finite-Mode Method (FM2) The central idea of FM2is to rebuild the Feynman path integral as the limit of a sequence of finite-dimensional oscillatory integrals. Instead of starting from a Euclidean functional measure and analytically continuing back to real time, we remain strictly on the Lorentzian axis and regularize the path integral by projecting the field onto a carefully chosen time–frequency frame. In this section we first explain why the standard plane-wave (Fourier) basis is mathematically ill-suited for real-time curved backgrounds, then construct the Gabor frame, derive the finitemode quadratic action, and finally show how the resulting Fresnel integrals are made convergent via Lefschetz thimbles. 2.1 Failure of the Pure Fourier Basis Let us denote spacetime coordinates by x= (t, x) and consider a real scalar field ϕ(x) on a time-dependent background such as an FRW spacetime. Formally, one may attempt to expand ϕ(t, x) = Zdω d3k (2π)4˜ ϕ(ω, k)ei(ωt−k·x),(3) and rewrite the action in terms of the Fourier coefficients ˜ ϕ(ω, k). However, this purely Fourierbased description suffers from three structural problems on a time-dependent background gµν(t, x): 1. Non-vanishing boundary terms. Plane waves eiωt do not decay as |t| → ∞. When deriving the quadratic form of the action, integrations by parts in tgenerically produce boundary terms of the form hap(t)¯ f(t)∂tg(t)i+∞ t=−∞,(4) where a(t) is the scale factor, pis a power determined by the dimension, and f, g are mode functions. With plane waves this term does not vanish and has to be discarded by hand (e.g., “switching off” the interaction in the distant past/future or enclosing the universe in a box), which is conceptually and mathematically unsatisfactory. 3 2. Lack of time localization. Fourier modes are global in time: a single mode eiωt extends across the entire cosmic history. In an FRW universe, the scale factor a(t) encodes strong time dependence near the Big Bang and in late-time acceleration. A global basis cannot isolate the local geometric features of a given epoch, making it difficult to control the short-distance / short-time structure of the propagator in a covariant way. 3. Dense action matrix. For a non-stationary background, different Fourier modes are strongly coupled. The quadratic form of the action in Fourier space, S[˜ ϕ] = 1 2Zdω dω′d3k d3k′˜ ϕ(ω, k)∗A(ω, k;ω′,k′)˜ ϕ(ω′,k′),(5) involves a kernel Athat is generically dense. This makes the spectral analysis of the operator ˆ Kand the evaluation of det Aextremely cumbersome, both conceptually and numerically. These shortcomings motivate the introduction of a time–frequency localized frame that decays in tand yields a sparse, structured action matrix. This is precisely the role played by the Gabor frame in FM2. 2.2 The Gabor Frame and Field Expansion We define the (1+1)-dimensional Gabor atom as gj(t, x) = g(m,n,k)(t, x) = exp−(t−tm)2 2σ2 | {z } Gaussian window eiωnt |{z} time modulation eikx |{z} spatial carrier ,(6) where tmis the time center, ωnthe central frequency, kthe spatial momentum, and σis the time–window width. For definiteness we first work in (1+1) dimensions; generalization to higher dimensions is straightforward. The key properties of this frame are: •Time–frequency localization: each gjis localized around (tm, ωn) with Gaussian decay in t. •Approximate completeness: the collection {gj}j∈Z3forms a (redundant) Gabor frame of L2(R2); any square-integrable field can be expanded as ϕ(t, x) = lim N→∞ N X j=−N ujgj(t, x), uj∈C,(7) with convergence in the L2norm. •Automatic vanishing of boundary terms: because of the Gaussian window, all mode functions and their derivatives vanish exponentially as |t| → ∞, so that surface terms from integration by parts vanish identically: [ap(t) ¯gm(t, x)∂tgn(t, x)]+∞ t=−∞ = 0.(8) In practice, FM2proceeds by truncating the expansion to a finite number Nof Gabor modes, ϕN(t, x) = N X j=1 ujgj(t, x),(9) and then taking N→ ∞ at the end of the calculation. This is the origin of the term finite-mode. 4 2.3 From the Action to a Finite-Dimensional Quadratic Form To make the construction concrete, let us work with a real scalar field on a (1+1)-dimensional FRW background, ds2=−dt2+a2(t)dx2,√−g=a(t),(10) with action S[ϕ] = 1 2Zdt dx a(t)−(∂tϕ)2+1 a2(t)(∂xϕ)2−m2ϕ2.(11) Substituting the finite-mode expansion ϕN(t, x) into (11) and using linearity, we obtain SN(u) = 1 2 N X m,n=1 ¯um(AN)mn un=1 2u†ANu,(12) where the matrix elements are (AN)mn =Zdt dx a(t)−(∂tgm) (∂tgn) + 1 a2(t)(∂xgm) (∂xgn)−m2gmgn.(13) Given the explicit form (6), the derivatives are ∂tgj(t, x) = −t−tm σ2+iωngj(t, x)≡Dt,j(t)gj(t, x),(14) ∂xgj(t, x) = ik gj(t, x).(15) Inserting these, the matrix naturally decomposes into three physical blocks: (AN)mn =Kmn +Gmn +Mmn,(16) with Kmn =−Zdt dx a(t)Dt,m(t)Dt,n(t) ¯gmgn,(kinetic block),(17) Gmn =Zdt dx a(t) a2(t)(ikm)(ikn) ¯gmgn=−Zdt dx 1 a(t)kmkn¯gmgn,(gradient block),(18) Mmn =−Zdt dx a(t)m2¯gmgn,(mass block).(19) Because of the Gaussian time window and the oscillatory spatial factor, all these integrals are convergent for each finite N. At this stage, the continuum path integral Z=ZDϕ e i ℏS[ϕ](20) has been rigorously reduced to a finite-dimensional Fresnel integral over the complex coefficients {uj}: ZN=ZR2N dN(Re u)dN(Im u) expi 2ℏu†ANu.(21) The FM2prescription is to evaluate ZNexactly for each finite Nand then study the limit N→ ∞. 5 2.4 Lefschetz Thimble Regularization of the Fresnel Integral The integral (21) is highly oscillatory because ANinherits the Lorentzian signature of the underlying spacetime: it has both positive and negative eigenvalues. Naively applying the flatspace Gaussian formula would be ill-defined. FM2overcomes this by applying Picard–Lefschetz theory mode by mode. Let λkand v(k)be the eigenvalues and eigenvectors of AN, so that in the eigenbasis the quadratic form is diagonal, SN(u) = 1 2 N X k=1 λk|yk|2, yk= (v(k))†u.(22) The integral factorizes: ZN= N Y k=1 ZR2 d2ykexpi 2ℏλk|yk|2.(23) Each factor is a two-dimensional Fresnel integral. For a given eigenvalue λkwe deform the integration contour into the complex plane along the steepest descent direction (the Lefschetz thimble): •If λk>0 (“matter-like” mode), we rotate the contour by +45◦: yk=eiπ/4xk, xk∈R, so that i 2λky2 k→ −1 2λkx2 k, and the integral becomes a convergent real Gaussian. •If λk<0 (“gravity / conformal” mode), we rotate by −45◦: yk=e−iπ/4xk, xk∈R, leading again to a negative-definite quadratic form in xk. In both cases the integral yields ZJk d2ykexpi 2ℏλk|yk|2=2πiℏ λk ,(24) up to an unimportant overall phase common to all modes. Taking the product over kand using Qkλk= det AN, we obtain the exact closed-form expression ZN=NNs(2πiℏ)N det AN ,(25) where NNis a phase factor independent of the background geometry. 6 2.5 Reconstruction of the Full Path Integral Finally, the FM2definition of the full Lorentzian path integral is Z[g] = lim N→∞ZN[g] = lim N→∞NNs(2πiℏ)N det AN[g].(26) Here AN[g] is the finite-mode Galerkin projection of the covariant kinetic operator ˆ Kg=−□g+m2,(27) onto the Gabor frame adapted to the FRW background. The effective action is then Γeff[g] = −iln Z[g] = i 2lim N→∞ Tr ln AN[g] + const.,(28) which is the object we analyze in later sections via heat-kernel and spectral techniques to derive the Casimir energy, the Mehler kernel, and ultimately the induced Einstein–Hilbert term. This completes the finite-mode reconstruction of the real-time Feynman path integral in a time-dependent FRW background. 3 Optical Analogy: Effective Metric as Refractive Index We now demonstrate that the spacetime metric can be reconstructed from the singularity structure of the propagator derived via FM2. In the high-frequency limit (N→ ∞), the propagator Gis the inverse of the kinetic operator. In Fourier space (ω, k), utilizing the coefficients from the ANmatrix construction in an FRW background, ˜ G(ω, k)∼1 −a3(t)ω2+a(t)k2=1 a(t)−a2(t)ω2+k2.(29) The pole structure defines the dispersion relation a2(t)ω2=k2,(30) implying a phase velocity vp=ω/|k|= 1/a(t). Viewing the vacuum as an optical medium, the effective refractive index is n(t) = c vp =a(t).(31) Performing the Fourier transform back to position space in 3 + 1 dimensions, G(x, x′)∼Zeik·(x−x′) k2d4k∼1 σ2(x, x′),(32) where σ2(x, x′)≃ −a2(t)∆t2+∆x2is the squared geodesic distance defined by the FRW metric. This 1/σ2behavior precisely matches the Hadamard singularity condition required for QFT in curved spacetime. Thus, FM2reconstructs the light-cone structure of General Relativity from the principal symbol of the kinetic operator. 4 Verification I: The Quantum Harmonic Oscillator We now use FM2to derive the propagator of the harmonic oscillator and verify that the determinant limit reproduces the exact Mehler kernel. 7 4.1 The Gel’fand–Yaglom Theorem Consider the interacting operator ˆ OV=−∂2 t−ω2(33) and the free operator ˆ O0=−∂2 t,(34) both defined on paths x(t) with fixed endpoints at t= 0 and t=T. The Gel’fand–Yaglom theorem states that the ratio of functional determinants is given by lim N→∞ det ˆ OV det ˆ O0 =ψV(T) ψ0(T),(35) where ψ(t) satisfies the homogeneous equation ˆ Oψ(t) = 0, ψ(0) = 0,˙ ψ(0) = 1.(36) 4.2 FM2Determinant Limit and Mehler Kernel In the FM2formalism, we project the operator onto a finite Gabor subspace: A(V) N=PNˆ OVPN, A(0) N=PNˆ O0PN,(37) so that the finite-mode partition functions read Z(V) N∝1 qdet A(V) N , Z(0) N∝1 qdet A(0) N .(38) Taking N→ ∞ and using the Gel’fand–Yaglom theorem, det A(V) N det A(0) N−−−−→ N→∞ det ˆ OV det ˆ O0 =ψV(T) ψ0(T).(39) For the harmonic oscillator, ¨ ψV+ω2ψV= 0 ⇒ψV(t) = sin(ωt) ω, ψV(T) = sin(ωT) ω,(40) while for the free particle, ¨ ψ0= 0 ⇒ψ0(t)=t, ψ0(T) = T. (41) Thus R≡det ˆ OV det ˆ O0 =sin(ωT) ωT .(42) The propagator prefactor is inversely proportional to the square root of the determinant. Normalizing by the known free-particle prefactor Kfree(T) = rm 2πiT ,(43) we obtain KHO(T) = Kfree(T)R−1/2=rm 2πiT sωT sin(ωT)=rmω 2πi sin(ωT),(44) which is precisely the Mehler kernel. This confirms that the FM2determinant limit reproduces the exact quantum dynamics of the oscillator in real time. 8 5 Verification II: (1+1)D Casimir Effect from Matrix Trace We next show how the FM2matrix trace yields the standard (1 + 1)-dimensional Casimir energy ECas =−π 24L(45) without invoking ad-hoc regularization such as ζ-function tricks. 5.1 FM2Matrix Construction and Energy Definition Consider a massless scalar field confined to an interval x∈[0, L] with Dirichlet boundary conditions ϕ(0) = ϕ(L) = 0. The spatial eigenmodes are kn=nπ L, n = 1,2, . . . (46) In FM2, the vacuum energy is obtained from the imaginary part of the effective action, Evac =−1 TIm Γeff ∼1 2X n ωnf(ωn/ΛGabor),(47) where ωn=knand fis a smooth cutoff function determined by the Gabor frame, satisfying f(0) = 1, f(ω)→0 rapidly as ω→ ∞.(48) The physical Casimir energy is defined as the difference between the energy in the presence of plates (discrete spectrum) and the reference energy of free space (continuous spectrum): ECas = lim Λ→∞ 1 2"∞ X n=1 F(n)−Z∞ 0 F(n)dn#, F(n)≡ωnf(ωn/Λ),(49) with ωn=nπ/L. 5.2 Euler–Maclaurin Expansion The difference between the sum and the integral can be evaluated using the Euler–Maclaurin formula: ∞ X n=0 F(n)−Z∞ 0 F(n)dn =1 2F(0) −B2 2! F′(0) + B4 4! F(3)(0) −··· ,(50) where B2= 1/6 is the second Bernoulli number. In our case, F(n) = nπ Lfnπ LΛ.(51) Near n= 0, the cutoff satisfies f(0) = 1 and f′(0) = 0, so F(0) = 0, F′(0) = π Lf(0) = π L.(52) The higher derivatives F(k)(0) involve derivatives of fand are suppressed by powers of 1/Λ; they either vanish as Λ → ∞ or contribute only L-independent constants that can be absorbed into a bulk renormalization. Keeping only the universal finite piece, ∞ X n=0 F(n)−Z∞ 0 F(n)dn ≃ −B2 2F′(0) = −1 12 π L.(53) 9