Toward quantization of inhomogeneous field theory

Abstract

We explore the quantization of a $(1+1)$-dimensional inhomogeneous scalar field theory in which Poincaré symmetry is explicitly broken. We show the ‘classical equivalence’ between a scalar field theory on curved spacetime background and its corresponding inhomogeneous scalar field theory. This implies that a hidden connection may exist among some inhomogeneous field theories, which corresponds to general covariance in field theory on curved spacetime. Based on the classical equivalence, we propose how to quantize a specific field theory with broken Poincaré symmetry inspired by standard field theoretic approaches, canonical and algebraic methods, on curved spacetime. Consequently, we show that the Unruh effect can be realized in inhomogeneous field theory and propose that it may be tested by a condensed matter experiment. We suggest that an algebraic approach is appropriate for the quantization of a generic inhomogeneous field theory.

Appendices

Appendix

A Isometries in $(1+1)$-dimensional background

In this Appendix, we summarize some formulae used in the main text. For the two-dimensional metric,

$$\begin{aligned} ds^2=e^{2\omega (x)}(-dt^2+dx^2). \end{aligned}$$

(71)

the non-vanishing Christoffel symbols are given by

$$\begin{aligned} \Gamma ^{t}_{tx}= \Gamma ^{t}_{xt}= \Gamma ^{x}_{xx} = \Gamma ^{x}_{tt}= \omega '(x), \end{aligned}$$

(72)

where ${}'$ denotes the differentiation with respect to x. In this geometry, the Ricci tensor and the curvature scalar are given by

$$\begin{aligned} R_{\mu \nu } = -g_{\mu \nu }\, \omega ''. \qquad R=-2e^{-2\omega }\omega ''. \end{aligned}$$

(73)

Now we show that the spacetime described by the above metric admits only a time-like Killing vector, excepting dS, AdS, and Minkowski spacetimes which have three Killing vectors. Killing condition on this background $\nabla _{(\mu } \xi _{\nu )}=0$ becomes

$$\begin{aligned} \xi ^{x} = C(t) e^{-\omega (x)}, \qquad (\xi ^{x})^{\varvec{\cdot }}-(\xi ^{t})' =0, \qquad (\xi ^{t})^{\varvec{\cdot }} + \omega ' \xi ^{x}= (\xi ^{t})^{\varvec{\cdot }} - (\xi ^{x})' =0, \end{aligned}$$

(74)

where $\varvec{\cdot }$ denotes the differentiation with respect to t. This condition leads to

$$\begin{aligned} \frac{{\ddot{C}}}{C} = \frac{(e^{-2\omega })''}{e^{-2\omega }} = A_{0} = \text{const}. \quad \text {if} \quad C\ne 0. \end{aligned}$$

(75)

When $C=0$, one obtains $\xi ^{\mu } = (1,0)$ or $\xi = \partial _{t}$, up to normalization. In the case of $C\ne 0$, we can solve the differential equation $\frac{(e^{-2\omega })''}{e^{-2\omega }} = - A_{0}$, which leads to

$$\begin{aligned} e^{-\omega } = \left\{ \begin{array}{lll } D_{c}\cosh Bx + D_{s}\sinh Bx, &{} \text {when} &{} A_{0} = B^{2} > 0 \\ D\cos B(x-x_{0}), &{} \text {when} &{} A_{0} = - B^{2} < 0 \\ D_{1}x + D_{2}, &{} \text {when} &{} A_{0} = 0 \end{array} \right. , \end{aligned}$$

(76)

where $D_{c/s}, D, D_{1/2}$, and $x_{0}$ are integration constants. One can check that upper two cases ($A_{0}$ is positive or negative) correspond to $\hbox {dS}_{{2}}$ and $\hbox {AdS}_{{2}}$, respectively. While the last one with $D_{1}=0$ corresponds to the Minkowski spacetime. If $D_{1}\ne 0$, then there is a singularity. This computation tells us that there is no other Killing vector except for $\xi = \partial _{t}$ ($C=0$ case) for a generic non-singular metric.

As an example, let us consider the case of $A_{0} =B^{2}$ with $D_{c}=0$. The independent Killing vectors up to normalization are obtained as

$$\begin{aligned} \xi = \xi ^{t}\partial _{t} + \xi ^{x}\partial _{\xi } = \left\{ \begin{array}{l} \cosh Bx\sinh Bt\, \partial _{t} + \sinh Bx \cosh Bt\, \partial _{x} \\ \cosh Bx\cosh Bt\, \partial _{t} + \sinh Bx \sinh Bt\, \partial _{x} \\ \partial _{t} \end{array} \right. . \end{aligned}$$

(77)

One can check that these three Killing vectors form a SO(2, 1) algebra. Indeed, performing the coordinate transformation $r=r_{H}\coth Bx$, we obtain the Rindler wedge of $\hbox {AdS}_{{2}}$ geometry. In the case of $A_{0} =B^{2}$ with $D_{s}=0$, one obtains

$$\begin{aligned} \xi = \xi ^{t}\partial _{t} + \xi ^{x}\partial _{\xi } = \left\{ \begin{array}{l} \sinh Bx\sinh Bt\, \partial _{t} + \cosh Bx \cosh Bt\, \partial _{x} \\ \sinh Bx\cosh Bt\, \partial _{t} + \cosh Bx \sinh Bt\, \partial _{x} \\ \partial _{t} \end{array} \right. , \end{aligned}$$

(78)

which corresponds to the Killing vectors in the static path of $\hbox {dS}_{{2}}$ spacetime.

B canonical quantization in curved spacetime

In this Appendix, we review the quantization procedure in a $(1+1)$-dimensional curved background [47, 64]. We consider the quadratic action (6) for the real scalar field with the coupling $\xi$ on a background geometry. The equation of motion of the model (6) is given by

$$\begin{aligned} \left( -\Box + m_0^2 + \xi {\mathcal {R}} \right) \phi = 0, \qquad \Box \equiv \frac{1}{\sqrt{-g}} \partial _{\mu }\left( \sqrt{-g} g^{\mu \nu } \partial _{\nu }\right) . \end{aligned}$$

(79)

The canonical momentum for the field $\phi$ at a constant time t is read as

$$\begin{aligned} \pi (t,x) \equiv \frac{\delta S_{\textrm{FTCS}}}{\delta \partial _{t} \phi (t,x)} = \sqrt{h} \, n^\mu \partial _{\mu } \phi (t,x), \end{aligned}$$

(80)

where $n^\mu = -\frac{g^{\mu 0}}{\sqrt{-g^{00}}}$ is the unit normal vector to the hypersurface $\Sigma$, and h is the determinant of the induced spatial metric on the surface. In order to quantize the field $\phi (t,x)$, one has to promote the fields, $\phi (t,x)$ and $\pi (t,x)$, to Hermitian operators and require the commutation relation at a fixed time t,

$$\begin{aligned}&{[} \phi (t,x), \pi (t, y) ] = i\delta (x-y). \nonumber \\&{[} \phi (t,x), \phi (t, y) ] = [ \pi (t,x), \pi (t, y) ]=0. \end{aligned}$$

(81)

The commutation relations in (81) are the same forms of those in the Minkowski spacetime. Following the quantization procedure in the Minkowski spacetime, we define an inner product

$$\begin{aligned} \langle \phi _1,\, \phi _2 \rangle = \int _\Sigma d \Sigma ^\mu J_\mu , \end{aligned}$$

(82)

where $d \Sigma ^\mu \equiv n^\mu d \Sigma$ and $J_\mu \equiv i \left( \phi _1^* \partial _{\mu } \phi _2 - \partial _{\mu } \phi _1^* \phi _2 \right)$ is a current satisfying the on-shell relation $\nabla ^\mu J_\mu = 0$. This bracket is called the Klein-Gordon inner product, and it does not depend on the choice of the spacelike hypersurface $\Sigma$ in the case that the fields decay sufficiently fast at spatial infinity. That is, if we consider another hypersurface $\Sigma '$ at a different time $t'$, we have the relation

$$\begin{aligned} \int _\Sigma d \Sigma ^\mu J_\mu - \int _{\Sigma '} d \Sigma ^\mu J_\mu = \int _{M} d^2 x \sqrt{-g} \, \nabla ^\mu J_\mu = 0, \end{aligned}$$

(83)

where M is the manifold bounded by the hypersurfaces, $\Sigma$ and $\Sigma '$. This independence of the hypersurfaces realizes the time-independence of the inner product in the Minkowski spacetime. For complex functions f and g satisfying the equation of motion (79), the inner product satisfies the following relations,

$$\begin{aligned} \langle f,\, g\rangle ^* = - \langle f^*,\, g^* \rangle = \langle g,\, f\rangle , \end{aligned}$$

(84)

which implies $\langle f,\, f^*\rangle =0$. In analogy with the quantization of $\phi$ in Minkowski spacetime, we define the annihilation operator related to the function f in terms of the inner product,

$$\begin{aligned} a(f) = \langle f,\, \phi \rangle , \end{aligned}$$

(85)

which is independent of the hypersurface $\Sigma$. Using the properties of the inner product in (84) and the Hermiticity of the field operator $\phi$, we obtain the Hermitian conjugate of a(f), $a^\dagger (f) = - a (f^*)$. Using these relations and the commutation relations in (81), we obtain the following commutation relations

$$\begin{aligned} {[}a(f), a^\dagger (g)] = \langle f,\, g\rangle , \quad [a(f), a (g)] = -\langle f,\, g^*\rangle , \quad [a^\dagger (f), a^\dagger (g)] =- \langle f^*,\, g\rangle . \end{aligned}$$

(86)

When the complex solution f satisfies $\langle f,\, f\rangle = 1$, by setting $g = f$ in (86) one can easily see that the relations in (86) are nothing but commutation relations of number operators in harmonic oscillator. Therefore in order to quantize the scalar field $\phi$ in a curved background we have to find a complete orthonormal basis of solutions to (79) satisfying the inner product relations

$$\begin{aligned} \langle u_i,\, u_j\rangle = \delta _{ij}, \quad \langle u^*_i,\, u_j\rangle = 0, \quad \langle u_i^*,\, u_j^*\rangle = - \delta _{ij}, \end{aligned}$$

(87)

where corresponding annihilation and creation operators are denoted by $a_i$ and $a_i^\dagger$, respectively. Then one can expand the scalar field $\phi$ as

$$\begin{aligned} \phi (t,x) =\sum _{i} \left( a_i u_i + a_i^\dagger u_i^*\right) \end{aligned}$$

(88)

with the commutation relations $[a_i,\, a_j^\dagger ] = \delta _{ij}$. The Fock space can be constructed by these operators.