Lifshitz Formula in Casimir Physics

• Lifshitz formula is a spectral identity that quantifies Casimir forces between dielectric or metallic planes using reflection coefficients evaluated at imaginary frequencies.
• It decomposes electromagnetic fields into TE and TM modes through logarithmic expressions and is validated by renormalization, functional determinant, and spectral summation methods.
• Extensions include finite-temperature and dispersive media formulations, resolving controversies in material response modeling and setting benchmarks for experimental Casimir pressure comparisons.

Searching arXiv for recent and canonical uses of “Lifshitz formula” across the main literatures represented in the source material. The Lifshitz formula, in standard Casimir usage, gives the Casimir energy or free energy per unit area between two parallel planar media with frequency-dependent electromagnetic response, expressed through reflection coefficients evaluated at imaginary frequencies. In its canonical form it treats two dielectric or metallic half-spaces separated by a gap, decomposes the electromagnetic field into TE and TM polarizations, and rewrites the interaction in terms of logarithms of multiple-reflection factors. The same surname also labels distinct formulas in other branches of mathematical physics, condensed matter, and holography, but the dominant technical meaning is the Casimir–Lifshitz formula for dispersive media (Miralaei et al., 2018, Milton et al., 2010).

1. Canonical Casimir–Lifshitz form

For two parallel dielectric half-spaces separated by a gap of width $a$, the zero-temperature Lifshitz energy per unit area can be written as

$$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$

with imaginary frequency $\omega=i\xi$, transverse wavevector $k_\perp$, longitudinal wavevectors

$$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$

and Fresnel coefficients

$$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$

The associated pressure is obtained from

$$\mathcal{F}(a) = -\frac{\partial E(a)}{\partial a}.$$

This reflection-coefficient representation is the standard zero-temperature Lifshitz formula for planar media (Miralaei et al., 2018).

At finite temperature, the same structure appears in Matsubara form,

$$\mathcal{F}(L,T) = \frac{k_B T}{2\pi} \sum_{n=0}^{\infty}{}' \int_0^{\infty} k_\perp \, dk_\perp \, \sum_{\text{pol}= \text{TE,TM}} \ln\!\Big(1 - r_{1,\text{pol}}\, r_{2,\text{pol}}\, e^{-2 \kappa_n L}\Big),$$

where $\xi_n = 2\pi n k_B T/\hbar$ are bosonic Matsubara frequencies and the prime indicates half weight for the $n=0$ term (Fujii et al., 2024). In the zero-temperature limit used in high-precision comparison with metallic plates, the pressure may be written as

$$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$0

with

$$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$1

and

$$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$2

where

$$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$3

This is the form used in direct pressure calculations for metallic plates (Geyer et al., 2010).

Domain	Core structure	Representative paper
Casimir physics	$$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$4 for TE/TM modes	(Miralaei et al., 2018)
Finite-temperature Lifshitz theory	Matsubara sum over imaginary frequencies	(Fujii et al., 2024)
Spectral summation form	Mode sum plus scattering density shift	(Nesterenko et al., 2011)
Functional-determinant form	$$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$5 of reduced operators	(Ttira et al., 2011)

2. Derivational frameworks

Several mathematically distinct routes lead to the same reflection-coefficient formula. One route uses box renormalization: the system is embedded in a large but finite box, the vacuum energy is computed with the plates at finite separation and at infinite separation, and the Casimir energy is defined as

$$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$6

Because both configurations are treated in the same finite box, bulk and surface self-energy divergences cancel, leaving the interaction energy in the limit $$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$7. After the argument principle and rotation to imaginary frequency, this yields the standard zero-temperature Lifshitz energy (Miralaei et al., 2018).

A second route uses functional determinants and the Gelfand–Yaglom theorem. For a real scalar field in $$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$8 dimensions interacting with two imperfect thick plane mirrors, the vacuum energy is reduced to a determinant ratio of one-dimensional operators in the normal direction, and Gelfand–Yaglom converts that ratio into boundary-value data. The resulting force per unit area takes the Lifshitz form

$$E(a) = \frac{\hbar}{2\pi^2} \int_0^\infty d\xi \int_0^\infty k_\perp\, dk_\perp \sum_{p=\text{TE, TM}} \ln\Bigl(1 - r_p^{(1)}(i\xi, k_\perp)\, r_p^{(2)}(i\xi, k_\perp)\, e^{-2k_3 a}\Bigr),$$9

with $\omega=i\xi$0 and reflection coefficients evaluated at imaginary frequencies. In this formulation the microscopic potential of each mirror is compressed into its Euclidean reflection coefficient (Ttira et al., 2011).

A third route uses spectral summation. In the planar electromagnetic problem, the spectrum is decomposed into surface modes, waveguide modes, and photonic modes, and the continuum contribution is written through the spectral density shift

$\omega=i\xi$1

The principal novelty of that approach is a special choice of contour passes in the complex $\omega=i\xi$2-plane that keeps track of the cuts connecting the branch points and the complex roots of the frequency equations. After contour deformation one arrives at

$\omega=i\xi$3

with

$\omega=i\xi$4

which is the zero-temperature Lifshitz formula for the symmetric three-layer problem (Nesterenko et al., 2011).

A fourth route begins from the dispersive electromagnetic energy itself. For linear media with $\omega=i\xi$5, the energy density contains the dispersive factor $\omega=i\xi$6,

$\omega=i\xi$7

Using the Green’s dyadic, this leads to a trace–log representation and then to the standard Lifshitz energy. In this account, the dispersive term is not optional: it is required for consistency with the trace–log formalism and for the variational identity used in Lifshitz-type derivations (Milton et al., 2010).

The same determinant logic extends to thin electromagnetic mirrors described not by bulk dielectric functions but by vacuum polarization tensors $\omega=i\xi$8. In that setting the electromagnetic problem splits into two scalar-like sectors built from invariant projectors, and in the zero-width limit the free energy becomes

$\omega=i\xi$9

with effective reflection coefficients

$k_\perp$0

This provides a microscopic QFT realization of Lifshitz theory for zero-width mirrors and graphene-like sheets (Fosco et al., 2012).

3. Material response, low-frequency modeling, and controversy

The central material input in the Lifshitz formula is $k_\perp$1. For gold, a common low-frequency extrapolation is the Drude model,

$k_\perp$2

while the generalized plasma-like model uses

$k_\perp$3

The choice between these prescriptions controls the low-frequency TE and TM reflection coefficients and therefore the thermal correction and entropy behavior (Geyer et al., 2010).

High-precision comparison with experiment has made this issue central. In detailed comparison with Casimir-pressure measurements between Au plates, the zero-temperature Lifshitz formula combined with Drude extrapolation using $k_\perp$4 and $k_\perp$5 was reported to be excluded at a 70% confidence level, while the zero-temperature Lifshitz formula combined with the generalized plasma-like model with $k_\perp$6 was reported to be experimentally consistent (Geyer et al., 2010).

The controversy is not only empirical. One line of analysis argues that inserting the Drude permittivity

$k_\perp$7

into the standard Matsubara formulation introduces a contribution that is non-perturbative in the relaxation parameter $k_\perp$8. In that analysis, the problematic term appears in the complex-frequency plane as a self-intersection or bifurcation of the integration path, and it is identified as the source of the nonzero entropy at $k_\perp$9 that violates the Nernst theorem (Bordag, 2011).

A different theoretical assessment reaches a more cautious conclusion. Using equilibrium two-time Green functions and an explicit derivation of the fluctuation–dissipation theorem, it has been argued that there are no substantial theoretical arguments in favour of using either plasma model permittivity or Drude model permittivity in the Lifshitz formula, and that the decision rests with comparison of theoretical calculations with experiment. In that account, the fluctuation–dissipation theorem remains applicable even when dissipation is absent, so explicit assertions to the contrary are rejected (Nesterenko, 2021).

The historical derivation itself has also been scrutinized. One recent critique argues that in Lifshitz’s own work the formula is given without a consistent conclusion, and that the approach in that work does not allow one to obtain it. On that reading, the Levin–Rytov, Schwinger variational, and Van Kampen methods provide more satisfactory derivational frameworks, especially for plane-layered structures (Davidovich, 12 Jan 2026).

4. Generalizations within Casimir theory

The reflection-coefficient structure of the Lifshitz formula extends well beyond two dielectric half-spaces. For Dirac fields at finite chemical potential, a finite-density Lifshitz formula was proposed in which the chemical potential appears as a shift of the imaginary frequency. For periodic boundary conditions in $$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$0, the Casimir energy per unit area becomes

$$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$1

with

$$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$2

In this formulation, finite density is not an external correction to Lifshitz theory but a direct modification of the spectral variable entering the logarithm (Fujii et al., 2024).

For a rectangular dielectric waveguide, the same logic leads to a generalized Lifshitz formula built from surface modes and mode-matching at the waveguide boundaries. The zero-point energy is written as an integral over longitudinal momentum $$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$3, imaginary frequency $$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$4, and a discrete transverse index $$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$5, with logarithms of generalized reflection-like factors. In the asymptotic limit the result recovers the classical expressions for perfect reflecting boundaries, and in the limit $$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$6 it reproduces the planar Lifshitz result (Arias et al., 2024).

For thin or zero-width mirrors, the vacuum polarization of the material can replace bulk $$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$7 altogether. This is the setting used for graphene sheets, where the zero-width limit identifies effective reflection amplitudes directly from the polarization invariants $$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$8 and $$k_3 = \sqrt{k_\perp^2 + \varepsilon_3(i\xi)\,\frac{\xi^2}{c^2}},\qquad k_j = \sqrt{k_\perp^2 + \varepsilon_j(i\xi)\,\frac{\xi^2}{c^2}},$$9. In that case,

$$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$0

$$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$1

and the Lifshitz structure persists in terms of those effective couplings rather than conventional Fresnel coefficients (Fosco et al., 2012).

These extensions suggest that the core content of the Lifshitz formula is not a specific dielectric model but a general spectral identity: once the response of each interface can be encoded into reflection data or an equivalent determinant, the interaction energy acquires the characteristic logarithmic multiple-reflection form.

5. Limiting regimes and asymptotics

The ideal-conductor limit remains the benchmark. For perfect conductors in vacuum, the zero-temperature pressure is

$$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$2

which is the reference case from which real-material Lifshitz theory departs (Davidovich, 12 Jan 2026). For planar dielectrics, the Lifshitz result approaches this form only when the relevant reflection coefficients tend to unit magnitude over the frequencies that dominate the integral.

At large separations, the retarded regime reproduces the familiar inverse fourth-power dependence on distance. At small gaps, however, one recent analysis reports that the force density changes the inverse fourth-degree dependence on the distance and practically ceases to depend on it at distances less than $$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$3 nm. In the same treatment, for thin identical plates the pressure density is proportional to the square of their thickness at such distances, but the dependence quickly becomes saturated and already at thicknesses of the order of $$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$4 nm practically ceases to depend on it (Davidovich, 12 Jan 2026).

Waveguide and finite-cross-section geometries introduce further limiting cases. In the rectangular dielectric waveguide, taking one transverse dimension to infinity converts the discrete transverse sum into a continuous integral and yields the planar Lifshitz formula. Taking the external permittivity to infinity recovers the perfect-conductor rectangular-waveguide Casimir energy, written there in terms of an Epstein zeta function (Arias et al., 2024). These transitions illustrate that the Lifshitz formula is not tied to a single geometry but to a hierarchy of limiting spectral problems.

6. Other formulas bearing the Lifshitz name

In other arXiv literatures, “Lifshitz formula” refers to formulas that are not part of Casimir electrodynamics. In asymptotically Lifshitz holography, the relevant relation is an energy–entropy law for universal-horizon black holes in Hořava–Lifshitz gravity. For $$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$5-dimensional asymptotically Lifshitz universal-horizon black holes, the paper reports

$$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$6

and identifies it with the field-theory expectation

$$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$7

This is presented there as the thermodynamic “Lifshitz formula” associated with anisotropic scaling exponent $$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$8 (Cheyne et al., 2021).

In quantum oscillation theory, the relevant object is the Lifshitz–Kosevich formula. For a two-band inverted insulator, the oscillatory part of the grand potential is written as

$$r_\text{TE}^{(j)}(i\xi, k_\perp) = \frac{k_3 - k_j}{k_3 + k_j},\qquad r_\text{TM}^{(j)}(i\xi, k_\perp) = \frac{\varepsilon_j(i\xi)\, k_3 - \varepsilon_3(i\xi)\, k_j} {\varepsilon_j(i\xi)\, k_3 + \varepsilon_3(i\xi)\, k_j}.$$9

with a modified amplitude function that reduces in suitable limits to the standard Lifshitz–Kosevich factor

$$\mathcal{F}(a) = -\frac{\partial E(a)}{\partial a}.$$0

Here the Lifshitz name enters through the de Haas–van Alphen and Shubnikov–de Haas oscillation formalism rather than Casimir theory (Grubinskas et al., 2017).

In operator theory, the relevant object is the Lifshitz–Kreĭn trace formula. For negative or nonpositive operators on Banach spaces, a spectral shift function is defined by

$$\mathcal{F}(a) = -\frac{\partial E(a)}{\partial a}.$$1

and the trace formula takes the contour-integral form

$$\mathcal{F}(a) = -\frac{\partial E(a)}{\partial a}.$$2

This is a perturbation-theoretic trace identity in Hirsch functional calculus, unrelated to Casimir forces except for the common surname and the shared use of logarithmic spectral data (Mirotin, 2018).

Taken together, these usages indicate that “Lifshitz formula” is not a single universal identity but a family of formulas associated with Evgeny Lifshitz’s influence across statistical physics, condensed matter, field theory, and mathematical physics. In the standard and historically dominant sense, however, the term denotes the Casimir–Lifshitz formula: the reflection-based expression for electromagnetic fluctuation forces between planar media (Miralaei et al., 2018).