---
title: Proper Time Expansion in Physics
url: https://www.emergentmind.com/topics/proper-time-expansion
type: topic
---

# Proper Time Expansion in Physics

Proper time expansion denotes several related analytic constructions in which fields, observables, kernels, or flow equations are organized in powers of a proper-time parameter. In relativistic heavy-ion theory the variable is the physical proper time $\tau$ in the forward light-cone; in heat-kernel and effective-action methods it is the Schwinger–DeWitt parameter $s$ or $T$; in real-time pair production it is a contour variable on the real axis; and in boost-invariant plasma dynamics a large-$\tau$ proper-time expansion becomes a transseries rather than a single power series [2304.03241] [1906.04019] [2111.06204] [1912.08135] [1810.07130].

## 1. Meanings of the proper-time parameter

In the glasma problem one works in covariant coordinates $x^\pm=(t\pm z)/\sqrt{2}$ and $x_\perp$, with the proper time
$$
\tau=\sqrt{2x^+x^-},
$$
or equivalently in Milne coordinates $x^\mu=(\tau,\eta,x,y)$ with $\tau=\sqrt{t^2-z^2}$ and $\eta=\tfrac12\ln(x^+/x^-)$; the forward light-cone description is usually paired with Fock–Schwinger gauge $A_\tau=0$ [2304.03241]. In heat-kernel theory, by contrast, the kernel is written as
$$
K(x,y;T)=\langle x|e^{-TA}|y\rangle,
$$
and $T$ is interpreted as the “proper-time” entering the heat equation $(\partial_T+A_x)K(x,y;T)=0$ [1906.04019].

A distinct usage appears in cosmology, where the Robertson–Walker line element
$$
ds^2=-c^2d\tau^2+a(\tau)^2\bigl[d\chi^2+\sin^2\chi(d\theta^2+\sin^2\theta\,d\phi^2)\bigr]
$$
identifies $\tau$ as the proper time measured by any comoving clock [1002.2932]. This suggests that “proper time” is not a single technical object but either a physical invariant time coordinate or an auxiliary parameterization of operators, depending on the framework.

## 2. Small-$\tau$ expansion in classical Yang–Mills glasma

For the earliest post-collision stage of relativistic heavy-ion collisions, the post-collision gauge fields are written in the forward light-cone as
$$
A^+(x)=\Theta(x^+)\Theta(x^-)\,x^+\,\alpha(\tau,x_\perp),\qquad
A^-(x)=-\Theta(x^+)\Theta(x^-)\,x^-\,\alpha(\tau,x_\perp),
$$
$$
A_i(x)=\Theta(x^+)\Theta(x^-)\,\alpha^\perp_i(\tau,x_\perp)+\text{pre-collision terms},
$$
and the post-collision fields are expanded in even powers of $\tau$,
$$
\alpha(\tau,x_\perp)=\sum_{n=0}^\infty \alpha^{(n)}(x_\perp)\tau^n,\qquad
\alpha^\perp_i(\tau,x_\perp)=\sum_{n=0}^\infty \alpha^{\perp(n)}_i(x_\perp)\tau^n.
$$
Through order $\tau^8$ all nonzero coefficients are even in $\tau$, and the energy–momentum tensor is likewise analytic at early times,
$$
T^{\mu\nu}(\tau,x_\perp)=\sum_{n=0}^\infty T^{\mu\nu(n)}(x_\perp)\tau^n.
$$
The forward-cone Yang–Mills equation is sourceless,
$$
D_\mu F^{\mu\nu}=0,
$$
with boundary conditions
$$
\alpha^\perp_i(0)=\beta_{1i}+\beta_{2i},\qquad
\alpha(0)=-\frac{ig}{2}[\beta_{1i},\beta_{2i}],
$$
so that each order in $\tau$ is determined by a chain of linear equations of the schematic form
$$
L\,\alpha^{(n)}(x_\perp)=\mathrm{Source}[\alpha^{(<n)},\partial_\perp\alpha^{(<n)}].
$$
At each $n$ one inverts trivial Laplacians or covariant derivatives on the 2-dimensional transverse plane [2304.03241].

For the energy density and longitudinal and transverse pressures,
$$
\epsilon(\tau)=T^{00}(\tau),\qquad P_T(\tau)=T^{xx}(\tau)=T^{yy}(\tau),\qquad P_L(\tau)=T^{zz}(\tau),
$$
the source-averaged series through $\tau^8$ read
$$
\epsilon(\tau)=\epsilon^{(0)}+\epsilon^{(2)}\tau^2+\epsilon^{(4)}\tau^4+\epsilon^{(6)}\tau^6+\epsilon^{(8)}\tau^8+O(\tau^{10}),
$$
$$
P_L(\tau)=-\epsilon^{(0)}+P_L^{(2)}\tau^2+P_L^{(4)}\tau^4+P_L^{(6)}\tau^6+P_L^{(8)}\tau^8+O(\tau^{10}),
$$
$$
P_T(\tau)=\epsilon^{(0)}+P_T^{(2)}\tau^2+P_T^{(4)}\tau^4+P_T^{(6)}\tau^6+P_T^{(8)}\tau^8+O(\tau^{10}).
$$
The displayed coefficient structures are nested commutators and covariant derivatives of the initial longitudinal electric and magnetic fields. For typical MV initial conditions at central impact parameter and $Q_s\tau\lesssim0.6$,
$$
\epsilon(\tau)\simeq \epsilon^{(0)}\bigl[1-0.21(Q_s\tau)^2+0.015(Q_s\tau)^4-0.0006(Q_s\tau)^6+1.2\times10^{-5}(Q_s\tau)^8\bigr],
$$
$$
P_L(\tau)\simeq -\epsilon^{(0)}\bigl[1-0.27(Q_s\tau)^2+0.019(Q_s\tau)^4-0.0008(Q_s\tau)^6+1.6\times10^{-5}(Q_s\tau)^8\bigr],
$$
$$
P_T(\tau)\simeq \epsilon^{(0)}\bigl[1-0.11(Q_s\tau)^2+0.005(Q_s\tau)^4-0.0002(Q_s\tau)^6+4\times10^{-6}(Q_s\tau)^8\bigr].
$$
These numbers illustrate that $P_L$ starts negative and approaches zero from below, while $\epsilon$ and $P_T$ remain positive.

## 3. Inhomogeneity, convergence, and extension of the early-time regime

A realistic nucleus has a finite transverse profile $\mu(x_\perp)$, for example a Woods–Saxon projection. In this setting the two-point correlator of pre-collision transverse potentials depends on the center-of-mass coordinate $R=(x_\perp+y_\perp)/2$,
$$
\langle \beta^a_{ni}(x)\beta^b_{nj}(y)\rangle
=\delta^{nn}\delta^{ab}\,B_{ij}(|x-y|;\mu(R)),
$$
and one expands $B_{ij}$ to first order in $\nabla_R\mu(R)$. This generates new terms such as
$$
\epsilon^{(n)}(R)=\epsilon^{(n)}_0[\mu(R)]+\epsilon^{(n)}_{\nabla^2\mu}\,\frac{\nabla^2\mu(R)}{Q_s^2}+\dots
$$
and similarly for $P_{T,L}^{(n)}$. These gradient corrections are formally $O(\nabla^2/Q_s^2)$ and numerically small in the interior of the nucleus ($R\lesssim4\,\mathrm{fm}$ for Pb) but grow near the edge. The same analysis gives an empirical radius of convergence: for observables derived from $T^{\mu\nu}$, the series through $O(\tau^8)$ remains reliable up to $\tau\simeq0.06\,\mathrm{fm}/c$ $(\approx0.6/Q_s)$, while jet quenching parameters computed through $O(\tau^7)$ converge out to $\tau\simeq0.08\,\mathrm{fm}/c$ [2304.03241].

A later study explored several methods to push the maximum reliable time upward in the classical Yang–Mills glasma calculation. For the key observable
$$
A_{TL}(\tau)=\frac{3(p_T-p_L)}{2p_T+p_L},
$$
evaluated at mid-rapidity and $R=b=0$, the reported limits are as follows [2510.06390].

| Method | $\tau_{\max}$ [fm/c] |
|---|---:|
| full YM (8th) | 0.05 |
| Li–Kapusta | 0.08 |
| Padé | 0.08 |
| ML | 0.065 |

The Li–Kapusta approximation assumes two well-separated hard scales, the classical cutoff $\Lambda$ and the saturation scale $Q_s$, with $\Lambda\gg Q_s\gg m$, expands $T^{\mu\nu}$ in powers of $Q_s^2/\Lambda^2$ and $m^2/\Lambda^2$, and by keeping only leading terms pushes the $\tau$-series to order 20. Padé extrapolation fits the full eighth-order series with low-order rational functions in $x=Q_s\tau$ and, for the energy density, extends convergence to $x\approx1.5$, corresponding to $\tau_{\max}\approx0.15\,\mathrm{fm}/c$. The machine-learning construction uses PYSR on synthetic eighth-order series data and the empirical ratio
$$
n_i/d_i=(9i/2)+6,\qquad i=0,2,4,\dots,8,
$$
obtaining $n_{10}=-2394\pm47$ and $n_{12}=+2606\pm76$, but systematic uncertainties limit reliable use to $\tau_{\max}\approx0.065\,\mathrm{fm}/c$. Taken together, these results retain the basic limitation of the method to the very earliest glasma stage, even when resummation or surrogate prediction is introduced.

## 4. Heat-kernel asymptotics and one-loop effective actions

The proper time method plays an important role in modern mathematics and physics. For a second-order elliptic operator
$$
A_x=-(D_x)^2-V(x),\qquad D_x=\partial_x+B(x),
$$
the heat kernel satisfies
$$
(\partial_T+A_x)K(x,y;T)=0,\qquad K(x,y;0^+)=\delta(x-y),
$$
and admits the standard short-time ansatz
$$
K(x,y;T)=(4\pi T)^{-d/2}\exp\!\left[-\frac{(x-y)^2}{4T}\right]\sum_{n=0}^\infty a_n(x,y)T^n.
$$
On the diagonal this becomes
$$
K(x,x;T)\sim(4\pi T)^{-d/2}\sum_{n=0}^\infty a_n(x)T^n.
$$
Substitution into the heat equation yields the transport equations
$$
(x-y)^\mu D_{x^\mu}a_0(x,y)=0,\qquad a_0(x,x)=I,
$$
$$
\bigl(n+(x-y)^\mu D_{x^\mu}\bigr)a_n(x,y)=-A_xa_{n-1}(x,y),\qquad n\ge1.
$$
A major simplification is achieved by choosing the Fock–Schwinger gauge about the base point $y$,
$$
(x-y)^\mu B_\mu(y+s(x-y))=0,
$$
which gives the explicit radial expansion
$$
B_\mu(y+X)=\int_0^1 ds\,s\,X^\nu F_{\nu\mu}(y+sX)\equiv f_\mu^{FS}(X;y),
$$
and a straight-line parallel transporter
$$
\Phi(x,y)=P\exp\Bigl\{-\int_y^x dz^\mu B_\mu(z)\Bigr\}.
$$
The diagonal coefficients can then be written in a closed, non-recursive combinatorial form; the first terms are
$$
a_0(x)=I,\qquad a_1(x)=V(x),\qquad
a_2(x)=V^2(x)+\tfrac16\nabla_\mu\nabla^\mu V(x)+\tfrac1{12}F_{\mu\nu}(x)F^{\mu\nu}(x).
$$
The same kernel also has a world-line representation,
$$
K(x,y;T)=\Phi(x,y)\int_{u(0)=y}^{u(T)=x}Du(s)\,
\exp\Bigl\{-\int_0^T ds\,[\tfrac14\dot u^2+V(u(s))]\Bigr\},
$$
and the coefficients obtained from the “exponential world-line” formula coincide order by order with those of the proper-time recurrence [1906.04019].

For one-loop effective actions with unequal masses, the Fock–Schwinger proper-time method starts from
$$
W_E\equiv-\ln|\det D_E|=-\frac12\int_0^\infty\frac{ds}{s}\,\mathrm{Tr}\,e^{-sA},
$$
with
$$
A\equiv D_E^\dagger D_E=M^2-d^2+Y,
$$
where $M=\mathrm{diag}(m_1,m_2,\dots,m_f)$ is diagonal in flavor space. Since $M$ does not commute with $A\equiv-d^2+Y$, one uses the factorization
$$
e^{-s(M^2+A)}=e^{-sM^2}\,O_E[-A](s)
=e^{-sM^2}\sum_{n=0}^\infty(-1)^n f_n(s;A),
$$
with nested proper-time integrals in $f_n$. The coincidence-limit coefficients are
$$
b_0(x,x)=\sum_{i=1}^f e^{-sm_i^2}E_i,
$$
$$
b_1(x,x)=
-\sum_i e^{-sm_i^2}Y_{ii}
-2\sum_{i<j}\frac{e^{-sm_i^2}-e^{-sm_j^2}}{m_j^2-m_i^2}\,r_a^{\,ij}T_a.
$$
After proper-time integration with the Pauli–Villars regulator
$$
P_\Lambda(s)=1-(1+s\Lambda^2)e^{-s\Lambda^2},
$$
one obtains
$$
a_0(x)=\sum_i E_i,
$$
$$
a_1(x)=
-\sum_i J_0(m_i^2)Y_{ii}
-\sum_{i<j}A\,J_0(m_i^2,m_j^2)\,r_a^{\,ij}T_a,
$$
which in the equal-mass limit collapse to the familiar $a_0=1$ and $a_1=-Y$ [2110.05590].

## 5. Exact proper-time Wilsonian renormalization-group flows

In the Wilsonian setting, proper time enters through a UV- and IR-regulating cutoff profile $f_k(s)$ for the scale-dependent action $S_k[\varphi]$. Requiring the partition function to be independent of the sliding UV scale $k$ leads to the exact flow equation
$$
\partial_t S_k[\varphi]
=\frac12\,\mathrm{Tr}\int_0^\infty\frac{ds}{s}\,
\bigl(\partial_t f_k(s)\bigr)e^{-sS_k^{(2)}[\varphi]},
$$
where $t=\ln k$ and $S_k^{(2)}$ is the Hessian in field space. A convenient one-parameter family of regulators is
$$
f_k(s;m)=
\frac{\gamma(m,msk^2)-\gamma(m,ms\Lambda^2)}{\Gamma(m)},
\qquad m>0,
$$
with
$$
\partial_t f_k(s;m)=r_k(s;m)=\frac{2}{\Gamma(m)}(msk^2)^m e^{-msk^2}.
$$
After the proper-time integration one arrives at the compact form
$$
\partial_t S_k[\varphi]
=
\mathrm{Tr}\,
\Bigl(\frac{mk^2}{S_k^{(2)}[\varphi]+mk^2}\Bigr)^m.
$$
As $m\to\infty$ this becomes the “exponential” flow $\partial_t S=(\mathrm{Tr}\,e^{-S^{(2)}/k^2})$, and two flows with parameters $m$ and $m'$ differ only by a field redefinition $\varphi(x)\mapsto \varphi(x)+\delta t\,\psi_x[\varphi]$. The parameter $m$ therefore labels equivalent coarse-graining schemes related by infinitesimal field redefinitions [1912.08135].

A derivative expansion to order $\partial^2$ uses
$$
S_k[\varphi]=\int d^dx\,\Bigl\{
V_k(\varphi)+\tfrac12 Z_k(\varphi)(\partial_\mu\varphi)(\partial^\mu\varphi)+\mathcal O(\partial^4)
\Bigr\},
$$
which produces $\beta$-functionals for the dimensionless potential $v(x)$ and wave-function renormalization $z(x)$. In $d=3$, fixed-point solutions of these equations describe the Ising (Wilson–Fisher) universality class. For the type-A regulator,
$$
\eta(m)\underset{m\to\infty}{\longrightarrow}0.0294,\qquad \nu\approx0.613,
$$
to be compared with the conformal-bootstrap values $\eta\approx0.0363$ and $\nu\approx0.630$. Preliminary explorations of spectrally adjusted schemes of type B and type C indicate even better convergence to the known critical exponents.

## 6. Real proper time, Padé singularities, and pair production

In scalar QED the real proper-time formulation writes the one-loop effective action in terms of the resolvent of
$$
H=\tfrac12\pi^2,\qquad \pi_\mu=p_\mu-eA_\mu(x),
$$
through
$$
\Gamma^\pm(m^2)=\int_0^{m^2}dz\,\mathrm{Tr}\,G^\pm(z)
=-i\int_{J_0^\pm}\frac{dt}{t}\,e^{\mp im^2t}\,\mathrm{Tr}\,e^{\mp iHt},
$$
with
$$
G^\pm(m^2)=(m^2+H\pm i0)^{-1},\qquad U^\pm(t)\equiv e^{\mp iHt}.
$$
The physical vacuum-vacuum amplitude is the gap
$$
\Delta\Gamma(m^2)\equiv\Gamma^+(m^2)-\Gamma^-(m^2)
=-i\int_{-\infty}^{\infty}\frac{dt}{t}\,e^{-im^2t}\,\mathrm{Tr}\,U(t),
$$
whose imaginary part gives the pair-production probability $P\approx2\,\mathrm{Im}\,\Gamma$ [2111.06204].

When $H$ cannot be diagonalized exactly, one splits $H=H_0+V$ and expands
$$
U(t)=\sum_{n=0}^\infty U_n(t),
$$
with recursion
$$
U_0(t)=e^{\mp itH_0},\qquad
U_n(t)=(-i)\int_0^t dt_1\,K(t,t_1)\,U_{n-1}(t_1).
$$
After expansion in the inhomogeneity parameter $\epsilon$ and field strength $eE$,
$$
\mathrm{Tr}\,U(t)=\sum_{\ell=0}^\infty \epsilon^\ell
\sum_{k=0}^\infty a_{\ell,k}\,t^{2k-2}(eE)^{2k}(\mathrm{derivatives})^\ell,
$$
each fixed-$\epsilon$ series converges only inside $|t|<R$. A diagonal or near-diagonal Padé approximant to the truncated series reveals the nonperturbative singularities
$$
t_\star=\frac{\pm2\pi i n}{eE},
$$
plus higher harmonics for the uniform electric field. The discontinuity of $\Delta\Gamma$ is then obtained from residues of poles in the upper or lower half-plane. For the first instanton,
$$
\mathrm{Im}\,\Delta\Gamma
=+V_4\frac{(eE)^2}{8\pi^3}e^{-2\pi m^2/(eE)}+\dots
$$
and in nonuniform backgrounds the order-$\epsilon^0$ term reproduces the locally constant field approximation
$$
\mathrm{Im}\,\Delta\Gamma_0
=\int d^4x\,\frac{[eE(x)]^2}{8\pi^3}
\exp\!\left[-\frac{2\pi m^2}{eE(x)}\right].
$$
For periodic backgrounds the classical limit produces WKB period integrals and the same actions can be computed along Lefschetz thimbles, yielding explicit agreement between the real proper-time construction and exact WKB. The unitarity of $P$ arises from the fact that the real-time proper-time contours $J_0^\pm$ prescribe unique analytic-continuation directions, so there is no Borel ambiguity in picking up pole residues.

## 7. Large proper time, hydrodynamic asymptotics, and transseries structure

Proper-time expansion also appears at late times. For boost-invariant $\mathcal N=4$ supersymmetric Yang–Mills plasma undergoing Bjorken flow, the energy density has a hydrodynamic gradient expansion
$$
\epsilon(\tau)=\sum_{k=0}^\infty a_k\,\tau^{-(4/3+2k/3)}
=\tau^{-4/3}\Bigl[\epsilon_0^{(0)}+\epsilon_1^{(0)}\tau^{-2/3}+\epsilon_2^{(0)}\tau^{-4/3}+\cdots\Bigr],
$$
with
$$
\epsilon_0^{(0)}=\pi^{-4},\qquad
\epsilon_1^{(0)}=-\frac23\pi^{-4},\qquad
\epsilon_2^{(0)}=\Bigl(\frac16+\frac{\ln2}{3}\Bigr)\pi^{-4}.
$$
However, the full late-time solution is not a single power series but a multi-parameter transseries whose exponentially suppressed sectors are in one-to-one correspondence with quasinormal modes of the dual AdS black brane [1810.07130].

Defining $u\equiv\tau^{2/3}$ and
$$
A_k\equiv \frac{3i}{2}\omega_k,
$$
the $k$th nonhydrodynamic sector has the structure
$$
\epsilon_k(\tau)\simeq
C_k\,\exp\!\Bigl[-\frac32 i\omega_k\tau^{2/3}\Bigr]\,
\tau^{-3\beta_k/2}\sum_{m=0}^\infty b_{k,m}\tau^{-2m/3},
$$
with
$$
\beta_k=-\frac{A_k}{6}+3.
$$
More generally,
$$
\epsilon(\tau,\{\sigma\})=
\sum_{n\in\mathbb N_0^\infty}
\sigma^n\,e^{-n\cdot A\,u}\,u^{-\beta_{(n)}}
\sum_{m=0}^\infty \epsilon_m^{(n)}u^{-m},
$$
where mixed sectors correspond to nonlinear couplings between quasinormal modes and
$$
\beta_{(n)}=-(n\cdot A)/6+n\cdot1+2.
$$
Borel singularities occur at $\xi=m\cdot A$, and the large-order behavior of the hydrodynamic coefficients is governed by these singularities. In this precise sense, the divergent hydrodynamic expansion already encodes the nonhydrodynamic sectors.

The comparison between early- and late-time constructions is structurally instructive. The small-$\tau$ glasma series is analytic in $\tau$ and has a finite empirical range of validity; the late-$\tau$ Bjorken expansion is asymptotic and requires transseries sectors and resurgence for completion. This suggests that “proper time expansion” is best understood not as a single algorithm but as a family of methods in which the choice of proper-time variable determines whether the relevant analytic object is a convergent short-time series, a gradient expansion, an exact flow equation, or a resurgent transseries.

Source: https://www.emergentmind.com/topics/proper-time-expansion