Scarf for Lifshitz

Abstract

Polarization of dispersive and dissipative dielectric media with smoothed-out inhomogeneities is studied with the goal to clarify the question of renormalizability of electromagnetic stress–energy tensor. The stress tensor is computed with the Lifshitz approach to van der Waals forces in the non-retarded limit, which accounts for dominant effects at the distances from the interface shorter than the absorption wavelength. After the substraction of the leading free space ultraviolet divergencies, there still remain two types of divergencies. First, contributions diverging in the sharp interface case become finite once it is smoothed out. Second, new subleading ultraviolet cut-off-dependent contributions appear due to the interface internal structure. The Hadamard expansion, based on the heat kernel method, is applied to systematically single out both finite and subleading contributions and to demonstrate incomplete renormalizability of the Lifshitz theory. The above approach also allows us to reveal the purely quantum mechanical nature of surface tension, which consists of finite cut-off-independent as well as cut-off-dependent contributions. The deduced theory of surface tension and its calculations for real dielectric media are favourably compared to the available experimental data. The problem of surface tension proves to be of a distinguished limit type because the sharp interface formulation loses the critical information about the internal structure of an interface. The general theory offered here is illustrated with an exactly solvable model representing a smooth transition between two different dielectric media, which relies upon a solution of the Schrödinger equation with the Scarf potential.

Heat kernel calculations

Let us define the new field

$$\begin{aligned} \begin{aligned} \phi =\mathrm {g}^{-1/4}\varPhi =\varepsilon ^{3/4}\varPhi \end{aligned}\end{aligned}$$

(78)

and the corresponding Green’s function

$$\begin{aligned} \begin{aligned} {\mathbb {G}}(\zeta ;{\varvec{x}},{\varvec{x'}})=\langle \upphi ({\varvec{x}})\upphi ({\varvec{x'}})\rangle =\mathrm {g}^{-1/4}({\varvec{x}})\,\mathrm {g}^{-1/4}({\varvec{x}}')\,{\widehat{G}}(\zeta ;{\varvec{x}},{\varvec{x'}}), \end{aligned}\end{aligned}$$

(79)

where $\upphi ({\varvec{x}})$ is the operator corresponding to the field $\phi ({\varvec{x}})$. Then, the inner product for this new field takes the covariant form $\int \mathrm {d}{\varvec{x}}\, \varPhi _1 \varPhi _2=\int \mathrm {d}{\varvec{x}}\,\varepsilon ^{-3/2}\, \phi _1 \phi _2=\int \mathrm {d}{\varvec{x}}\sqrt{\mathrm {g}}\, \phi _1 \phi _2$. Equation (3) transforms to

$$\begin{aligned} \begin{aligned} {\mathbb {O}} \, {\mathbb {G}}(\zeta ;{\varvec{x}},{\varvec{x'}})=\frac{\delta ({\varvec{x}}-{\varvec{x'}})}{\sqrt{\mathrm {g}}}, \ \text {where} \ {\mathbb {O}}=\mathrm {g}^{-1/4}({\varvec{x}})\,{\widehat{O}}\,\mathrm {g}^{1/4}({\varvec{x}}). \end{aligned}\end{aligned}$$

(80)

Taking into account that

$$\begin{aligned} \begin{aligned} \partial _j \varepsilon ^{-3/4}&=\varepsilon ^{-3/4}\Big (\partial _j-\frac{3}{4}\eta _j \Big )\\ \partial _i\partial _j \varepsilon ^{-3/4}&=\varepsilon ^{-3/4} \Big (\partial _i\partial _j-\frac{3}{4}\eta _i\partial _j -\frac{3}{4}\eta _j\partial _i-\frac{3}{4}(\partial _i\eta _j)+\frac{9}{16}\eta _i\eta _j \Big ), \end{aligned}\end{aligned}$$

(81)

the new operator ${\mathbb {O}}$ becomes

$$\begin{aligned} \begin{aligned} {\mathbb {O}}&=\varepsilon \delta ^{ij}\,\Big (\partial _i\partial _j-\frac{1}{2}\eta _i\partial _j\Big ) -\frac{3}{4}\varepsilon \delta ^{ij}\,\Big (\partial _i\eta _j+\frac{1}{4}\eta _i\eta _j \Big ). \end{aligned}\end{aligned}$$

(82)

Note that

$$\begin{aligned} \begin{aligned} \nabla _k\eta ^k=\varepsilon \delta ^{ij}\Big (\partial _i\eta _j-\frac{1}{2}\eta _i\eta _j\Big )=\mathrm {g}^{ij}\Big (\partial _i\eta _j-\frac{1}{2}\eta _i\eta _j\Big ) \end{aligned}\end{aligned}$$

(83)

and the covariant Laplacian in metric $\mathrm {g}_{ij}$ reads

$$\begin{aligned} \begin{aligned} \bigtriangleup&=\mathrm {g}^{ij}\nabla _i\nabla _j=\frac{1}{\sqrt{\mathrm {g}}}\partial _i\Big (\mathrm {g}^{ij}\sqrt{\mathrm {g}}\,\partial _j\Big ) =\mathrm {g}^{ij}\,\Big (\partial _i\partial _j-\frac{1}{2}\eta _i\partial _j \Big ). \end{aligned}\end{aligned}$$

(84)

Therefore, we obtain the covariant form for the operator in (80)

$$\begin{aligned} \begin{aligned} {\mathbb {O}}=\bigtriangleup -{\mathbb {V}}, \ \text {where} \ {\mathbb {V}}=\frac{3}{4}\Big (\nabla ^k\eta _k+\frac{3}{4}\eta ^k\eta _k\Big ) =\frac{3}{4}\varepsilon \delta ^{ij}\,\Big (\partial _i\eta _j+\frac{1}{4}\eta _i\eta _j \Big ), \end{aligned}\end{aligned}$$

(85)

which is Hermitian with the covariant measure $\sqrt{\mathrm {g}}$. The potential ${\mathbb {V}}$ can also be obtained following the computations in [31, 64], which require the knowledge of the Christoffel symbols

$$\begin{aligned} \varGamma ^i_{jk}=-\frac{1}{2}\big [\delta _{ij}\eta _k+\delta _{ik}\eta _j -\delta _{jk}\eta _i \big ], \ \mathrm {g}^{jk}\varGamma ^i_{jk}&=\frac{1}{2} \varepsilon \,\eta _i, \end{aligned}$$

(86)

and the connection

$$\begin{aligned} \begin{aligned} \omega _i=\frac{1}{2}\big [\eta _i+\mathrm {g}_{ik}\mathrm {g}^{mn}\varGamma ^k_{mn}\big ]=\frac{3}{4}\eta _i. \end{aligned}\end{aligned}$$

(87)

Heat kernel corresponding to operator ${\widehat{O}}$ is ${\widehat{K}}(s|{\varvec{x}},{\varvec{x'}})=\mathrm {g}^{1/4}({\varvec{x}})\mathrm {g}^{1/4}({\varvec{x}}'){\mathbb {K}}(s|{\varvec{x}},{\varvec{x'}})$, where

$$\begin{aligned} \begin{aligned} {\mathbb {K}}(s|{\varvec{x}},{\varvec{x}}')=-e^{s{\mathbb {O}}}=-\frac{\varDelta ^{1/2}({\varvec{x}},{\varvec{x}}')}{(4\pi s)^{3/2}}e^{-\frac{\sigma ({\varvec{x}},{\varvec{x}}')}{2s}}\Big (a_0({\varvec{x}},{\varvec{x}}')+s\,a_1({\varvec{x}},{\varvec{x}}')+\dots \Big ); \end{aligned}\end{aligned}$$

(88)

here s is the proper time, $\sigma ({\varvec{x}},{\varvec{x}}')$ a world function defined as one half of the square of geodesic distance between points in the metric $\mathrm {g}_{ij}$ and $\varDelta ^{1/2}({\varvec{x}},{\varvec{x}}') \equiv \mathrm {g}^{-1/2}({\varvec{x}}) \mathfrak {D}({\varvec{x}},{\varvec{x}}') \mathrm {g}^{-1/2}({\varvec{x}}')$, where $\mathfrak {D}({\varvec{x}},{\varvec{x}}') = -\det \left( -\sigma _{;\mu \nu '}\right)$ is the van Vleck–Morette determinant. For the computation of the heat kernel of this operator, we need to know $\varDelta ^{1/2}({\varvec{x}},{\varvec{x}}')$, $a_0({\varvec{x}},{\varvec{x}}')$, and $a_1({\varvec{x}},{\varvec{x}}')$ and their derivatives. In particular they include the curvature terms:

$$\begin{aligned} R_{ij}=\frac{1}{2}\partial _j\eta _i+\frac{1}{4}\eta _i \eta _j + \frac{1}{4} \delta _{ij}\delta ^{km}\big (2\partial _k\eta _m - \eta _k\eta _m\big ), \quad R = 2\varepsilon \,\delta ^{km}\Big (\partial _k\eta _m-\frac{1}{4}\eta _k\eta _m\Big ); \end{aligned}$$

(89)

here we used the relations

$$\begin{aligned}&\eta _{i;j}=\partial _j\eta _i+\eta _i\eta _j-\frac{1}{2}\delta _{ij}\delta ^{km}\eta _{k}\eta _m, \end{aligned}$$

(90a)

$$\begin{aligned} &\eta ^k{}_{;k}=\varepsilon \delta ^{ij} \big (\partial _j\eta _i-\frac{1}{2}\eta _{i}\eta _j), \partial _i (\mathrm {g}^{1/4})=-\frac{3}{4}\varepsilon ^{-3/4}\eta _i. \end{aligned}$$

(90b)

The regularized Green’s function (79), in the original notation, assumes the form

$$\begin{aligned} \begin{aligned} {\widehat{G}}_{\lambda _{0}}(\zeta ;{\varvec{x}},{\varvec{x'}})=\int _{\lambda _{0}^{2}}^{\infty } \mathrm {d}s\, {\widehat{K}}(s|{\varvec{x}},{\varvec{x}}') =\mathrm {g}^{1/4}({\varvec{x}})\mathrm {g}^{1/4}({\varvec{x}}')\int _{\lambda _{0}^{2}}^{\infty } \mathrm {d}s\,{\mathbb {K}}(s|{\varvec{x}},{\varvec{x'}}), \end{aligned}\end{aligned}$$

(91)

where $\lambda _{0}$ is the cut-off parameter, and can be expanded as

$$\begin{aligned} \begin{aligned} {\widehat{G}}_{\lambda _{0}}(\zeta ;{\varvec{x}},{\varvec{x'}})=-\int _{\lambda _{0}^{2}}^{\infty } \mathrm {d}s\, \frac{\mathfrak {D}^{1/2}({\varvec{x}},{\varvec{x}}')}{(4\pi s)^{3/2}}e^{-\frac{\sigma ({\varvec{x}},{\varvec{x}}')}{2s}}\Big (a_0({\varvec{x}},{\varvec{x}}')+s\,a_1({\varvec{x}},{\varvec{x}}')+\dots \Big ) \end{aligned}\end{aligned}$$

(92)

Derivation of the regularized stress tensor then reduces to the following steps:

• i) Compute $\varDelta ^{1/2}({\varvec{x}},{\varvec{x}}')$, $a_0({\varvec{x}},{\varvec{x}}')$, and $a_1({\varvec{x}},{\varvec{x}}')$;

• ii) Compute the second partial derivative of the Green’s function $\partial _{{\varvec{x}}}\partial _{{\varvec{x}}'}{\widehat{G}}_{\lambda _{0}}(\zeta ;{\varvec{x}},{\varvec{x'}})$;

• iii) Take the limit ${\varvec{x}}={\varvec{x'}}$;

• iv) Integrate over the proper time s and then expand the result in the cut-off $\lambda _{0}$;

• v) Compute ${\widehat{\sigma }}_{ij}(\zeta ;{\varvec{x}})$ by substituting the result to (11).

For the quantities in the limit of coincident points, we introduce notation $[\dots ]\equiv (\dots )|_{{\varvec{x}}={\varvec{x'}}}$. The expressions that we need to compute divergent parts (33) of the Green’s function ${\widehat{G}}_{\lambda _{0}}(\zeta ;{\varvec{x}},{\varvec{x'}})$ are [111]

$$\begin{aligned} {[}\sigma ]&=0, \quad [\sigma _{;i}]=0, \quad [\sigma _{;i'}]=0, \quad [\sigma _{;ij}]=\mathrm {g}_{ij}, [\sigma _{;ij'}]=-\mathrm {g}_{ij}, \end{aligned}$$

(93a)

$$\begin{aligned} {[}a_0]&=1, \quad [a_0{}_{;i}]=0, ~ [a_0{}_{;ij'}]=0, ~ [a_1]=\frac{1}{6}R-{\mathbb {V}}, ~ [a_1{}_{;i}]=\frac{1}{2}\big (\frac{1}{6}R_{;i}-{\mathbb {V}}_{;i}\big ), \end{aligned}$$

(93b)

$$\begin{aligned} {[}\varDelta ^{1/2}]&=1, \quad [(\varDelta ^{1/2})_{;i}]=0, \quad [(\varDelta ^{1/2})_{;ij}]=\frac{1}{6}R_{ij}, \quad [(\varDelta ^{1/2})_{;ij'}]=-\frac{1}{6}R_{ij}, \end{aligned}$$

(93c)

$$\begin{aligned}&[\partial _i\partial _{j'}\big (\mathrm {g}^{1/4}({\varvec{x}})\mathrm {g}^{1/4}({\varvec{x'}})\varDelta ^{1/2}\big )] =\varepsilon ^{-3/2}\left( \frac{9}{16}\eta _i\eta _j-\frac{1}{6}R_{ij}\right) . \end{aligned}$$

(93d)