Proper Time Expansion in Physics

• Proper Time Expansion is a framework that expresses fields, observables, and kernels as power series in a proper-time parameter across various physical contexts.
• It is used in settings ranging from relativistic heavy-ion collisions and heat-kernel methods to cosmology and Wilsonian renormalization-group flows.
• The method incorporates convergent series, Padé extrapolation, and transseries techniques to capture both early-time analytic regimes and late-time hydrodynamic asymptotics.

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 (Carrington et al., 2023, Ivanov et al., 2019, Pazarbaşı, 2021, Bonanno et al., 2019, Aniceto et al., 2018).

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 $s$0 (Carrington et al., 2023). In heat-kernel theory, by contrast, the kernel is written as

$s$1

and $s$2 is interpreted as the “proper-time” entering the heat equation $s$3 (Ivanov et al., 2019).

A distinct usage appears in cosmology, where the Robertson–Walker line element

$s$4

identifies $s$5 as the proper time measured by any comoving clock (Leffert, 2010). 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-$s$6 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

$s$7

$s$8

and the post-collision fields are expanded in even powers of $s$9,

$T$0

Through order $T$1 all nonzero coefficients are even in $T$2, and the energy–momentum tensor is likewise analytic at early times,

$T$3

The forward-cone Yang–Mills equation is sourceless,

$T$4

with boundary conditions

$T$5

so that each order in $T$6 is determined by a chain of linear equations of the schematic form

$T$7

At each $T$8 one inverts trivial Laplacians or covariant derivatives on the 2-dimensional transverse plane (Carrington et al., 2023).

For the energy density and longitudinal and transverse pressures,

$T$9

the source-averaged series through $\tau$0 read

$\tau$1

$\tau$2

$\tau$3

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 $\tau$4,

$\tau$5

$\tau$6

$\tau$7

These numbers illustrate that $\tau$8 starts negative and approaches zero from below, while $\tau$9 and $x^\pm=(t\pm z)/\sqrt{2}$0 remain positive.

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

A realistic nucleus has a finite transverse profile $x^\pm=(t\pm z)/\sqrt{2}$1, 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 $x^\pm=(t\pm z)/\sqrt{2}$2,

$x^\pm=(t\pm z)/\sqrt{2}$3

and one expands $x^\pm=(t\pm z)/\sqrt{2}$4 to first order in $x^\pm=(t\pm z)/\sqrt{2}$5. This generates new terms such as

$x^\pm=(t\pm z)/\sqrt{2}$6

and similarly for $x^\pm=(t\pm z)/\sqrt{2}$7. These gradient corrections are formally $x^\pm=(t\pm z)/\sqrt{2}$8 and numerically small in the interior of the nucleus ($x^\pm=(t\pm z)/\sqrt{2}$9 for Pb) but grow near the edge. The same analysis gives an empirical radius of convergence: for observables derived from $x_\perp$0, the series through $x_\perp$1 remains reliable up to $x_\perp$2 $x_\perp$3, while jet quenching parameters computed through $x_\perp$4 converge out to $x_\perp$5 (Carrington et al., 2023).

A later study explored several methods to push the maximum reliable time upward in the classical Yang–Mills glasma calculation. For the key observable

$x_\perp$6

evaluated at mid-rapidity and $x_\perp$7, the reported limits are as follows (Carrington et al., 7 Oct 2025).

Method	$x_\perp$8 [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 $x_\perp$9 and the saturation scale $\tau=\sqrt{2x^+x^-},$0, with $\tau=\sqrt{2x^+x^-},$1, expands $\tau=\sqrt{2x^+x^-},$2 in powers of $\tau=\sqrt{2x^+x^-},$3 and $\tau=\sqrt{2x^+x^-},$4, and by keeping only leading terms pushes the $\tau=\sqrt{2x^+x^-},$5-series to order 20. Padé extrapolation fits the full eighth-order series with low-order rational functions in $\tau=\sqrt{2x^+x^-},$6 and, for the energy density, extends convergence to $\tau=\sqrt{2x^+x^-},$7, corresponding to $\tau=\sqrt{2x^+x^-},$8. The machine-learning construction uses PYSR on synthetic eighth-order series data and the empirical ratio

$\tau=\sqrt{2x^+x^-},$9

obtaining $x^\mu=(\tau,\eta,x,y)$0 and $x^\mu=(\tau,\eta,x,y)$1, but systematic uncertainties limit reliable use to $x^\mu=(\tau,\eta,x,y)$2. 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

$x^\mu=(\tau,\eta,x,y)$3

the heat kernel satisfies

$x^\mu=(\tau,\eta,x,y)$4

and admits the standard short-time ansatz

$x^\mu=(\tau,\eta,x,y)$5

On the diagonal this becomes

$x^\mu=(\tau,\eta,x,y)$6

Substitution into the heat equation yields the transport equations

$x^\mu=(\tau,\eta,x,y)$7

$x^\mu=(\tau,\eta,x,y)$8

A major simplification is achieved by choosing the Fock–Schwinger gauge about the base point $x^\mu=(\tau,\eta,x,y)$9,

$\tau=\sqrt{t^2-z^2}$0

which gives the explicit radial expansion

$\tau=\sqrt{t^2-z^2}$1

and a straight-line parallel transporter

$\tau=\sqrt{t^2-z^2}$2

The diagonal coefficients can then be written in a closed, non-recursive combinatorial form; the first terms are

$\tau=\sqrt{t^2-z^2}$3

The same kernel also has a world-line representation,

$\tau=\sqrt{t^2-z^2}$4

and the coefficients obtained from the “exponential world-line” formula coincide order by order with those of the proper-time recurrence (Ivanov et al., 2019).

For one-loop effective actions with unequal masses, the Fock–Schwinger proper-time method starts from

$\tau=\sqrt{t^2-z^2}$5

with

$\tau=\sqrt{t^2-z^2}$6

where $\tau=\sqrt{t^2-z^2}$7 is diagonal in flavor space. Since $\tau=\sqrt{t^2-z^2}$8 does not commute with $\tau=\sqrt{t^2-z^2}$9, one uses the factorization

$\eta=\tfrac12\ln(x^+/x^-)$0

with nested proper-time integrals in $\eta=\tfrac12\ln(x^+/x^-)$1. The coincidence-limit coefficients are

$\eta=\tfrac12\ln(x^+/x^-)$2

$\eta=\tfrac12\ln(x^+/x^-)$3

After proper-time integration with the Pauli–Villars regulator

$\eta=\tfrac12\ln(x^+/x^-)$4

one obtains

$\eta=\tfrac12\ln(x^+/x^-)$5

$\eta=\tfrac12\ln(x^+/x^-)$6

which in the equal-mass limit collapse to the familiar $\eta=\tfrac12\ln(x^+/x^-)$7 and $\eta=\tfrac12\ln(x^+/x^-)$8 (Osipov, 2021).

5. Exact proper-time Wilsonian renormalization-group flows

In the Wilsonian setting, proper time enters through a UV- and IR-regulating cutoff profile $\eta=\tfrac12\ln(x^+/x^-)$9 for the scale-dependent action $s$00. Requiring the partition function to be independent of the sliding UV scale $s$01 leads to the exact flow equation

$s$02

where $s$03 and $s$04 is the Hessian in field space. A convenient one-parameter family of regulators is

$s$05

with

$s$06

After the proper-time integration one arrives at the compact form

$s$07

As $s$08 this becomes the “exponential” flow $s$09, and two flows with parameters $s$10 and $s$11 differ only by a field redefinition $s$12. The parameter $s$13 therefore labels equivalent coarse-graining schemes related by infinitesimal field redefinitions (Bonanno et al., 2019).

A derivative expansion to order $s$14 uses

$s$15

which produces $s$16-functionals for the dimensionless potential $s$17 and wave-function renormalization $s$18. In $s$19, fixed-point solutions of these equations describe the Ising (Wilson–Fisher) universality class. For the type-A regulator,

$s$20

to be compared with the conformal-bootstrap values $s$21 and $s$22. 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

$s$23

through

$s$24

with

$s$25

The physical vacuum-vacuum amplitude is the gap

$s$26

whose imaginary part gives the pair-production probability $s$27 (Pazarbaşı, 2021).

When $s$28 cannot be diagonalized exactly, one splits $s$29 and expands

$s$30

with recursion

$s$31

After expansion in the inhomogeneity parameter $s$32 and field strength $s$33,

$s$34

each fixed-$s$35 series converges only inside $s$36. A diagonal or near-diagonal Padé approximant to the truncated series reveals the nonperturbative singularities

$s$37

plus higher harmonics for the uniform electric field. The discontinuity of $s$38 is then obtained from residues of poles in the upper or lower half-plane. For the first instanton,

$s$39

and in nonuniform backgrounds the order-$s$40 term reproduces the locally constant field approximation

$s$41

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 $s$42 arises from the fact that the real-time proper-time contours $s$43 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 $s$44 supersymmetric Yang–Mills plasma undergoing Bjorken flow, the energy density has a hydrodynamic gradient expansion

$s$45

with

$s$46

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 (Aniceto et al., 2018).

Defining $s$47 and

$s$48

the $s$49th nonhydrodynamic sector has the structure

$s$50

with

$s$51

More generally,

$s$52

where mixed sectors correspond to nonlinear couplings between quasinormal modes and

$s$53

Borel singularities occur at $s$54, 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-$s$55 glasma series is analytic in $s$56 and has a finite empirical range of validity; the late-$s$57 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.