# Quantisation of Klein–Gordon field in \(\kappa \) space-time: deformed oscillators and Unruh effect

## Abstract

In this paper we study the quantisation of scalar field theory in \(\kappa \)-deformed space-time. Using a quantisation scheme that use only field equations, we derive the quantisation rules for deformed scalar theory, starting from the \(\kappa \)-deformed equations of motion. This scheme allows two choices; (1) a deformed commutation relation between the field and its conjugate which leads to usual oscillator algebra, (2) an undeformed commutation relation between field and its conjugate leading to a deformed oscillator algebra. This deformed oscillator algebra is used to derive modification to Unruh effect in the \(\kappa \)-space-time.

## 1 Introduction

Non-commutative space-time was introduced long back as an approach to handle divergences in quantum field theory [1]. Efforts to construct a renormalisable quantum theory of gravity brought the non-commutative space-time into intense scrutiny in last decades [2]. Appearance of Moyal space-time, a specific type of non-commutative space-time, in the low energy limit of a string theory [3] fueled major activities in the area of non-commutative theories(for detailed surveys, see the reviews [4, 5]). Non-commutative space-time also emerged naturally in the discussions of different approaches to quantum gravity [6, 7].

The characteristic features of field theory on non-commutative space-time are their intrinsic non-locality and non-linearity. They also incorporate a minimal length, in an efficient way into the discussion. This notion of minimal length is a common feature of different approaches to microscopic theory of gravity, such as string theory [3], dynamical triangulation [8], asymptotically safe models, fuzzy physics [9, 10] etc. The field theories on non-commutative space-time also show a mixing of UV and IR divergences [11]. The analysis of symmetries of field theories on non-commutative space-time brought out the important role played by Hopf algebras [12, 13, 14].

Different aspects of field theory models defined on \(\kappa \)-space-time have been studied in recent times [18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In most of these studies one start with the deformed dispersion relation, compatible with the \(\kappa \)-Poincare algebra and arrive at the consistent field equations. These field equations involve higher derivatives terms, signaling the underlying non-local nature of these models. Lagrangians are constructed so as to lead to these field equations as Euler-Lagrange equations and hence involve higher derivatives. This makes the canonical quantisation of these models a problematic one. Gauge theories in \(\kappa \)-space-time were constructed and studied in [28, 29, 30].

The canonical quantisation procedure requires the knowledge of the explicit form of the Lagrangian, but there is an alternative quantisation method discussed in [31, 32], which does not require the Lagrangian, rather it uses the equations of motion as the starting point for quantisation. In [33] the massive spin one gauge field in commutative space-time was quantised using this procedure. This quantisation method, unlike the canonical scheme, does not use the notion of conjugate momentum. The first step here is to define unequal time commutation relation between the field and its adjoint such that the field equations are compatible with Heisenberg’s equations of motion. This, then leads to commutation relations between the creation and annihilation operators that appear in the mode expansion of fields. Apart from the quantisation, this procedure also provides an elagant way to construct conserved currents directly from the equation of motion, without any reference to Lagrangian. Using this method one can also obtain the currents corresponding to discrete symmetries [34].

The non-commutative field theories are non-local and non-linear, and they also posses higher derivative terms. So construction of Lagrangian and quantisation of such non-commutative field theories are not trivial. Hence the canonical quantisation for such non-commutative field theories are difficult. In [35] the quantisation of Klein–Gordon field in \(\kappa \)-deformed space-time was studied and shown that the compatibility between quantisation and action of twisted flip operator [36] leads to deformed oscillator algebra in \(\kappa \)-Minkowski space-time.

Equations satisfied by different field theory models in \(\kappa \)-deformed space-time were derived, with out any reference to Lagrangians in recent times. In [37], Maxwell’s equations in \(\kappa \)-space-time were derived using Feynman’s approach and in [38] the \(\kappa \)-deformed Maxwell’s equations in terms of fields defined in the commuttaive space-time were derived by elevating the principle of minimal coupling to non-commutative space-time. \(\kappa \)-deformed geodesic equation was derived by generalising the Feynman’s approach to \(\kappa \) space-time in [39]. \(\kappa \)-deformed Dirac equation was analysed in [40]. The approach used in this paper can be employed for quantising these models also.

In this paper, we adapt the quantisation scheme of [31, 32, 33] and apply it to \(\kappa \)-deformed scalar field equation, valid up to first order in the deformation parameter. This scalar field equation was derived using the quadratic Casimir of the undeformed \(\kappa \)-Poincare algebra [21]. This field equation contains infinitely many derivatives and in the commutative limit, reduced to the well known Klein–Gordon equation. There is another generalisation of \(\kappa \)-deformed Klein–Gordon equation, which also reduce to the corrrect commutative limit [21]. In this paper, we derive the unequal time commutation relation between the field and its adjoint, satisfying the modified Klein–Gordon equation. Then by appealing to the higher derivative theory methods [41, 42, 43], we introduce the conjugate momentum correspodning to the field and (1) derive the canonical commutation relation between them and show that this leads to the usual oscillator algebra between the creation and annihilation operators; (2) starting with a specific deformed algebra satisfied by the creation and annihilation operators, show that the field and conjugate momentum satisfy usual commutation relation as in the commutative space-time. We then investigate the modification to the Unruh effect [44, 45, 46] due to \(\kappa \)-deformation using this deformed oscillators. We show that the total number of particles seen by an accelerated observer in the Minkowski vacuum is modified due to the non-commutativity of the space-time, but the Unruh temperature is unaffected. In [47, 48, 49], Unruh effect in \(\kappa \)-space-time was analysed using different approaches and modifications to Unruh temperature was obtained.

Organisation of this paper is as follows, In Sect. 2 we briefly discuss the procedure involved in quantising a scalar field using its equation of motion. We also recall the construction of conserved current for free fields corresponding to the symmetry transformations using this formalism. In Sect. 3 we start with \(\kappa \)-space-time whose coordinates satisfy a Lie algebra type commutation relation given in Eq. (1.2) Using the quadratic Casimir of the undeformed \(\kappa \)-Poincare algebra we set up \(\kappa \)-deformed Klein–Gordon equation for a particular choice of realisation, valid upto first order in *a*. Now we follow the quantisation procedure [31, 32] outlined in Sect. 2 and quantise the Klein–Gordon theory living in \(\kappa \)-space-time. Here we find the unequal time commutation relation between the field and its adjoint, which is modified due to the space-time non-commutativity. In Sect. 3.1, we derive the modified commutation relation between deformed field and its conjugate momentum and show that the corresponding creation and annihilation operators satisfy the usual oscillator algebra. In Sect. 3.2 we take an alternate route. Here we propose a generic, deformed oscillator algebra for the deformed creation and annihilate operators and show that this leads to usual commutation relation between deformed field and its conjugate momentum as in the commutative space-time. Using this deformed oscillator algebra we find that the eigen values of deformed number operator get modified by a multiplicative factor. This factor is the same as the one appearing in the modification of the commutation relation between deformed creation and deformed annihilate operators. In Sect. 4 we derive modifications to Unruh effect due to \(\kappa \)-deformation, by expanding the \(\kappa \)-deformed Klein–Gordon field in \(1 + 1\) dimensions defined in Minkowski space-time and Rindler space-time, whose frequency modes are connected through Bogoliubov transformation. We show that the Unruh effect gets a modification due to the non-commutativity but the Unruh temperature is unaffected. In Sect. 5 we give our concluding remarks. In appendix we show the calculational details of the construction of deformed conjugate momentum using the formalism of higher derivative theories. Here we work with \(\eta _{\mu \nu }=\hbox {diag}(-1,+1,+1,+1)\).

## 2 Quantisation of Klein–Gordon field

In [32] Y. Takahashi and H. Umezawa have shown that the quantisation of the fields can be done using their equations of motion. The most interesting part of this procedure is that it does not employ the canonical Lagrange formalism for quantisation. Similarly the conserved currents can also be derived directly from the equations of motion [31, 33].

*F*[

*x*] and

*G*[

*x*] are some functionals of the field operator. Now under these transformations Eqs. (2.2) and (2.3) become

## 3 \(\kappa \)-deformed scalar theory and deformed oscillators

In this section we quantise the \(\kappa \)-deformed Klein–Gordon equation using the formulation discussed in Sect. 2 and obtain a deformed commutation relation between creation and annihilate operators. Using this we evaluate the deformed eigen values of creation, annihilate and number operators, defined in \(\kappa \)-space-time.

*a*as

*a*, using the method summarised in Sect. 2. We start by defining

*a*. This gives us two equations corresponding to

*a*independent and

*a*dependent coefficients given by

*a*, is written as

*a*. Hence the solution to the \(\kappa \)-deformed Klein–Gordon equation valid upto first order in

*a*is

*a*dependent correction term, which is expressed as \(\hat{\Delta }(x-x')=\Delta (x-x')+af(x-x')\). Hence for a field operator obeying \(\kappa \)-deformed real Klein–Gordon equation (\(\hat{{\bar{\phi }}}(x')\) becomes \(\hat{\phi }(x')\)), Eq. (3.23) becomes

*a*dependent terms on the RHS as \(f(x-y)\), which is giving

### 3.1 Case 1

In this subsection we derive deformed equal time commutation relation between scalar field and its conjugate momentum defined in \(\kappa \)-space-time.

*a*as

### 3.2 Case 2

In this subsection we derive an undeformed equal time commutation relation between field and its conjugate momentum by proposing a deformed commutator between creation and annihilate operators in \(\kappa \)-space-time.

*h*(

*a*) is an arbitrary linear function in

*a*, and lim \(a\rightarrow 0, h(a)=1\). Now we use this modified commutation relation in Eq. (3.25) and follow the above steps to get canonical equal time commutation relation between non-commutative field operator and its conjugate momentum as

*g*(

*a*) is an arbitrary linear function in

*a*, such that in the limit \(a\rightarrow 0,g(a)\) becomes 1. Now we use Eq. (3.34) and Eq. (3.38) in Eq. (3.35) and evaluate \(c_-\) to be

## 4 Deformed Unruh effect

In this section we use the deformed oscillator algebra given in Eq. (3.34) and study the modifications in Unruh effect due to \(\kappa \)-deformation.

A uniformly accelerating (with constant proper acceleration *A*) observer in Minkowski space-time, observes the particles in a thermal bath with temperature, \(T=\frac{A\hbar }{2\pi k}\) in the Minkowski vacuum and this is known as Unruh effect [44, 45, 46].

In [47] *a* dependent correction term to the Unruh effect was obtained by considering the interaction between detector and scalar field defined in \(\kappa \)-space-time. In [48] correction to Unruh effect, valid up to order \(a^2\) that comes from the response function of a uniformly accelerating detector coupled to \(\kappa \)-deformed Klein–Gordon field written in commutative space-time was discussed. Similarly in [49] *a* dependent correction to the Unruh effect was calculated by analysing the response function of a uniformly accelerating detector coupled to massless \(\kappa \)-deformed Dirac field.

Here we use an alternative approach used in the commutative space-time in calculating the Unruh efect. This method uses the Bogoliubov transformations, relating the creation and annihilation operators appearing in the mode expansion of fields in two different basis. Using this relation, one calculate the vacuum expectation value of the number operator defined in one basis(by accelerating observer), over the vacuum defined in the second basis (by a stationary observer). This expectation value is shown to be non-zero and in the commutative space-time, the value obtained shows the existence of particle in thermal bath with a temperature related to the constant acceleration of the observer, in the vacuuum defined by the stationary observer in the Minkowski space-time. We use this approach to calculate the modification to Unruh effect in the \(\kappa \)-deformed space-time.

We expand the \(\kappa \)-deformed scalar field in \(1+1\) dimensions defined in two different basis, namely Minkowski space-time and Rindler space-time. Then we use Bogoliubov transformation to connect the frequency modes of the left moving sectors of Minkowski space-time with that of the Rindler space-time. Now use this Bogoliubov coefficients and the deformed oscillator algebra to derive the \(\kappa \)-deformed modications to the Unruh effect.

*A*is a positive constant. Thus the Minkowski line element defined in Eq. (4.1) takes the form

*U*and

*V*becomes

*U*and

*V*become

*V*and we obtain,

*a*dependent correction in Eq. (4.36) vanishes, reducing it to the commutative value. In [47, 48, 49] also, corrections to the Unruh effect were obtained. But in these cases the \(\kappa \)-deformation leads to correction in Unruh temperature unlike in the present case. Since \(\hat{u}(x)\propto u_0(x)\) (see Eq. (3.20)) upto first order in

*a*, we note that the frequency,

*w*is not modified. This may not be true if corrections to all orders in

*a*are included in the calculation. Since f requency is not modified upto first order in

*a*, from Eq. (4.36) we see that Unruh temperature is unaffected.

## 5 Conclusions

In this paper we have quantised the \(\kappa \)-deformed scalar field, which staisfies deformed Klein–Gordon equation in the \(\kappa \)-space-time, for a particular choice of realisation. This equation is construced from the quadratic Casimir of the corresponding symmetry algebra, viz; undeformed \(\kappa \)-Poincare algebra.

In the \(\kappa \)-space-time, different possible generalisations of Klein–Gordon field equations, all of which reduced to correct commutative limit have been studied. Construction of field equation is guided by the conditions (1) that they all, in the limit of vanishing non-commutative parameter, give well established equations of commutative space-time, (2) they all should give same deformed energy-momentum relations valid for the non-commutative space-time. In the case of \(\kappa \)-deformed space-time, Maxwell’s equations [37, 38], geodesic equation [39] and Dirac equation [40] were constructed without any reference to Lagrangian.

Starting from the deformed equation of motion for the scalar field theory, after constructing the operators \(\Lambda (\partial )\), Klein–Gordon divisor and \(\Gamma _\mu \) operator, we have derived, deformed, unequal time commutation relation between field and its adjoint, valid upto first order in the deformation parameter *a*. We then adopted the definition of (deformed) conjugate momentum given by the formalism of higher derivative theories [41, 42, 43] and derived equal time commutation relations between the field and its conjugate momentum. In Sect. 3.1 we have derived a deformed commutation relation between scalar field and its conjugate momentum in \(\kappa \)-space-time, valid upto first order in *a*. This leads to usual harmonic oscillator algebra between the deformed creation and annihilate operators. In Sect. 3.2 we showed that it is possible to obtain deformed commutation relations between creation and annihilation operators. Using this set of deformed operators, we have constructed number basis and showed that under \(\kappa \)-deformation, eigen values of the creation, annihilate and number operators pick up a \((1-2a\alpha )\) factor. This factor is the same as the \(\kappa \)-deformed modification present in the deformed oscillator algebra. Similar modification in the eigen values were seen in the context of *q*-deformed oscillators also [50].

In [35] quantisation of the scalar field satisfying \(\kappa \)-deformed Klein–Gordon equation given by \((\Box -m^2)\hat{\phi }(x)=0\), was studied. The compatibility between the flip operator and the modified product rule resulted in the modification to the commutation relation between scalar fields defined at two different points. These modified commutation relation was used to derive the deformed oscillator algebra. In the present case product rule is not modified under \(\kappa \)-deformation, but the unequal time commutation relation between field and its adjoint is modified. This resulted in the deformed oscillator algebra (which is different from one obtained in [35]).

We used the deformed oscillator algebra derived to study the modification in the Unruh effect, due to \(\kappa \)-deformation of the space-time. We employed the approach of Bogoliubov transformations in deriving the modification to Unruh effect. We showed that the vacuum expectation value of the deformed number operator defined by the Rindler observer gets modified by a factor which is same as the one that appeared in the deformed oscillator algebra. Interestingly, this modification does not alter the Unruh temperature and it remains unchanged under \(\kappa \)-deformation.

## Notes

### Acknowledgements

EH thanks Prof. V. Srinivasan for introducing to the reference [31]. EH thanks SERB, Govt. of India, for support through EMR/2015/000622. VR thanks Govt. of India, for support through DST-INSPIRE/IF170622.

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