Lagrangian Formulation of an Infinite Derivative Real Scalar Field Theory in the Framework of the Covariant Kempf-Mangano Algebra in a $(D+1)$-dimensional Minkowski Space-time

TL;DR

This paper develops a covariant infinite derivative scalar field theory based on Kempf-Mangano algebra, revealing a connection to non-local theories and identifying a characteristic length scale near nuclear dimensions.

Contribution

It introduces a novel infinite derivative scalar field formulation using covariant Kempf-Mangano algebra, linking it to non-local theories and low-energy Pais-Uhlenbeck oscillator behavior.

Findings

The theory describes two bosonic particles, including a ghostlike particle.

At low energies, the model behaves like a Pais-Uhlenbeck oscillator.

The characteristic length scale is close to nuclear dimensions (~10^{-15} m).

Abstract

In 2017, G. P. de Brito and co-workers suggested a covariant generalization of the Kempf-Mangano algebra in a $(D+1)$-dimensional Minkowski space-time [A. Kempf and G. Mangano, Phys. Rev. D \textbf{55}, 7909 (1997); G. P. de Brito, P. I. C. Caneda, Y. M. P. Gomes, J. T. Guaitolini Junior, and V. Nikoofard, Adv. High Energy Phys. \textbf{2017}, 4768341 (2017)]. It is shown that reformulation of a real scalar field theory from the viewpoint of the covariant Kempf-Mangano algebra leads to an infinite derivative Klein-Gordon wave equation which describes two bosonic particles in the free space (a usual particle and a ghostlike particle). We show that in the low-energy (large-distance) limit our infinite derivative scalar field theory behaves like a Pais-Uhlenbeck oscillator for a spatially homogeneous field configuration $\phi(t,\vec{\textbf{x}})=\phi(t)$. Our calculations show that there is a characteristic length scale $\delta$ in our model whose upper limit in a four-dimensional Minkowski space-time is close to the nuclear scalar, i.e., $\delta_{max}\sim \delta_{nuclear\ scale}\sim 10^{-15}\, m$. Finally, we show that there is an equivalence between a non-local real scalar field theory with a non-local form factor ${\cal K}(x-y)= -\frac{\square_x}{(1-\frac{\delta^2}{2}\square_x)^2} \ \delta^{(D+1)}(x-y)$ and an infinite derivative real scalar field theory from the viewpoint of the covariant Kempf-Mangano algebra.

Equations

$$S[x]={\int }_{t_i}^{t_f}dtL(x(t),\dot{x}(t)).$$

$$S[x]={\int }_{t_i}^{t_f}dtL(x^{(0)}(t),x^{(1)}(t),\dots ,x^{(N)}(t)),$$

$$\sum _{i=0}^N{\left(-\frac{d}{dt}\right)}^i\frac{\partial L}{\partial x^{(i)}}=0.$$

$$\begin{gathered} S[\varphi ]=\underset{kineticterm}{\underbrace{\frac{1}{2c}\int d^{D+1}x{d}^{D+1}y\varphi (x)\mathcal{K}(x-y)\varphi (y)}} \\ -\underset{localpotentialterm}{\underbrace{\frac{1}{c}\int d^{D+1}xV(\varphi (x))}}+\underset{interactionterm}{\underbrace{\frac{1}{c}\int d^{D+1}xJ(x)\varphi (x)}}, \end{gathered}$$

$$\mathcal{K}(x-y)=F({\square }_x){\delta }^{(D+1)}(x-y),$$

$$\Delta X\Delta P⩾\frac{\hslash }{2}\left[1+\beta (\Delta P{)}^2+\cdots \right],\beta >0.$$

$$\left[{\hat{X}}^i,{\hat{P}}^j\right]$$

$$\left[{\hat{X}}^i,{\hat{X}}^j\right]$$

$$\left[{\hat{P}}^i,{\hat{P}}^j\right]$$

$$\left[{\hat{X}}^{\mu },{\hat{P}}^{\nu }\right]$$

$$\left[{\hat{X}}^{\mu },{\hat{X}}^{\nu }\right]$$

$$\left[{\hat{P}}^{\mu },{\hat{P}}^{\nu }\right]$$

$${\hat{X}}^{\mu }$$

$${\hat{P}}^{\mu }$$

$${\hat{x}}^{\mu }⟶{\hat{X}}^{\mu }$$

$${\partial }_{\mu }⟶{\nabla }_{\mu }$$

$$\mathcal{L}(\varphi ,{\partial }_{\mu }\varphi )=\frac{1}{2}\left({\partial }_{\mu }\varphi (x)\right)\left({\partial }^{\mu }\varphi (x)\right)-\frac{1}{2}{\left(\frac{mc}{\hslash }\right)}^2{\varphi }^2(x).$$

$$\varphi (x)⟶\Phi (X)=\varphi (x).$$

$$\mathcal{L}$$

$$S[\varphi ]={\int }_{{\mathbb{R}}^{1,D}}{d}^{D}xdt\mathcal{L}(\varphi ,{\partial }_{{\nu }_1}\varphi ,{\partial }_{{\nu }_1}{\partial }_{{\nu }_2}\varphi ,{\partial }_{{\nu }_1}{\partial }_{{\nu }_2}{\partial }_{{\nu }_3}\varphi ,\cdots ).$$

$$\frac{\partial \mathcal{L}}{\partial \varphi }-{\left(\frac{\partial \mathcal{L}}{\partial {\varphi }_{{\mu }_1}}\right)}_{{\mu }_1}+{\left(\frac{\partial \mathcal{L}}{\partial {\varphi }_{{\mu }_1{\mu }_2}}\right)}_{{\mu }_1{\mu }_2}-\cdots +(-1{)}^k{\left(\frac{\partial \mathcal{L}}{\partial {\varphi }_{{\mu }_1{\mu }_2\cdots {\mu }_k}}\right)}_{{\mu }_1{\mu }_2\cdots {\mu }_k}+\cdots =0,$$

$${\varphi }_{{\mu }_1{\mu }_2\cdots {\mu }_k}:={\partial }_{{\mu }_1}{\partial }_{{\mu }_2}\cdots {\partial }_{{\mu }_k}\varphi ,$$

$$\frac{\partial {\varphi }_{{\mu }_1{\mu }_2\cdots {\mu }_k}}{\partial {\varphi }_{{\nu }_1{\nu }_2\cdots {\nu }_k}}={\delta }_{{\mu }_1}^{{\nu }_1}{\delta }_{{\mu }_2}^{{\nu }_2}\cdots {\delta }_{{\mu }_k}^{{\nu }_k}.$$

$$\frac{1}{{\left(1-\frac{(\hslash \ell {)}^2}{2}\square \right)}^2}\square \varphi (x)+{\left(\frac{mc}{\hslash }\right)}^2\varphi (x)=0.$$

$$\square \varphi (x)+{\left(\frac{mc}{\hslash }\right)}^2\varphi (x)+\underset{high-energy(short-distance)corrections}{\underbrace{(\hslash \ell {)}^2\square \square \varphi (x)+\frac{3}{4}(\hslash \ell {)}^4\square \square \square \varphi (x)+\mathcal{O}\left((\hslash \ell {)}^6\right)}}=0.$$

$$\varphi (x)=A{e}^{-\frac{i}{\hslash }\mathbf{p}.\mathbf{x}},$$

$$\frac{{\mathbf{p}}^2}{(1+\frac{{\ell }^2}{2}{\mathbf{p}}^2{)}^2}=m^2{c}^2.$$

$$E_{+}^2(\ell )$$

$$E_{-}^2(\ell )$$

$$m_{+}(\ell )$$

$$m_{-}(\ell )$$

$$\ell ⩽\frac{1}{\sqrt{2}}\frac{1}{mc}.$$

$${\delta }_{max}=\frac{1}{\sqrt{2}}\frac{\hslash }{mc}.$$

Full text

Keywords: Classical field theories; Relativistic wave equations; Nonlinear or nonlocal theories and models; Higher derivatives; Canonical formalism, Lagrangians, and variational principles; Pais-Uhlenbeck oscillator; Characteristic length scale

PACS: 03.50.-z, 03.65.Pm, 11.10.Lm, 04.20.Fy

1 Introduction

In classical mechanics, the motion of a point particle is described by the following action functional

[TABLE]

In 1850, the Russian mathematician Mikhail Ostrogradsky proposed a higher-order generalization of Lagrangian mechanics [1]. The action functional for a higher-order derivative mechanical system is

[TABLE]

where $x^{(i)}(t):=\frac{d^{i}x(t)}{dt^{i}}$ and $i\in\{0,1,\ldots,N\}$.

The variation of the above action functional with respect to $x$ leads to the following generalized Euler-Lagrange equation

[TABLE]

In 1942, B. Podolsky suggested a higher-order derivative generalization of Maxwell electrodynamics, in which the electrostatic self-energy of a point charge was a finite value [2].

Six years later, A. E. S. Green presented a higher-order derivative meson-field theory, in which the potential energy for a point nucleon at the origin was a finite value [3]. In 1950, A. Pais and G. E. Uhlenbeck showed that the appearance of higher-order derivative terms in the Lagrangian density of a quantum field theory can eliminate the ultraviolet divergences that appear in the $S$-matrix elements [4].

Today we know that the addition of higher-order derivative terms to the action functional of a quantum field theory is a possible way of regularizing quantum field theories [5-9]. Recently, the infinite derivative scalar field theory of the form

[TABLE]

has attracted a considerable attention because of its importance in non-local quantum field theory and string field theory [10].

The operator ${\cal K}(x-y)$ in Eq. (2) has the following general form [10]:

[TABLE]

where $F(\square_{x})$ is an entire analytic function of the d’Alembertian operator $\square_{x}$. On the other hand, different theories of quantum gravity such as string theory and loop quantum gravity predict the existence of a minimal length scale of the order of the Planck length [11-15]. Today we know that the existence of a minimal length scale in different theories of quantum gravity leads to a generalization of Heisenberg uncertainty principle as follows [12]:

[TABLE]

The above generalized uncertainty principle leads to a minimal length scale $\Delta X_{0}=\hbar\sqrt{\beta}\ (\Delta X_{0}>0)$ in the measurement of space intervals [12]. It should be noted that the reformulation of quantum field theory in the presence of a minimal length scale is another possible way for regularizing a quantum field theory [12]. The aim of this paper is reformulation of a real scalar field theory from the viewpoint of a covariant generalization of the Kempf-Mangano algebra which was proposed by G. P. de Brito and co-workers in Ref. [14].

This paper is organized as follows. In Section 2, a covariant generalization of the Kempf-Mangano algebra in a $(D+1)$-dimensional Minkowski space-time is presented briefly. In Section 3, we show that reformulation of a real scalar field theory in the framework of the covariant Kempf-Mangano algebra leads to a generalized Klein-Gordon wave equation with infinitely many derivatives. The free space solutions of this generalized Klein-Gordon wave equation describe two bosonic particles. In Section 4, we show that in the low-energy (large-distance) limit the infinite derivative scalar field theory which was formulated in Section 3 behaves like a Pais-Uhlenbeck oscillator for a spatially homogeneous field configuration $\phi(t,\vec{\textbf{x}})=\phi(t)$ [4].

Our calculations in Sections 3 and 4 together with numerical evaluations in Section 5 show that there is a characteristic length scale in our generalized real scalar field theory whose upper limit is very near to the nuclear scale, i.e., $10^{-15}\ m$. Finally, it should be emphasized that the results of this paper in the low-energy regime are compatible with the results of the standard Klein-Gordon theory. We use SI units in this paper. The flat space-time metric has the signature $\eta_{\mu\nu}=\eta^{\mu\nu}=diag(+,\underbrace{-,\ldots,-}_{D\ times}).$

2 The Covariant Kempf-Mangano Algebra

In 1997, A. Kempf together with G. Mangano proposed a one-parameter extension of the Heisenberg algebra [12]. The Kempf-Mangano algebra in a $D$-dimensional Euclidean space is described by the following generalized commutation relations:

[TABLE]

where $i,j=1,2,\ldots,D$ and $\ell$ is a non-negative constant parameter with dimension of $[momentum]^{-1}$ [12]. The reformulation of non-relativistic quantum mechanics and a charged scalar field in the framework of Kempf-Mangano algebra have been studied in details in Ref. [13].

In 2017, G. P. de Brito and co-workers proposed a covariant generalization of the Kempf-Mangano algebra [14]. The covariant Kempf-Mangano algebra in a $(D+1)$-dimensional Minkowski space-time is described by the following generalized commutation relations:111In 2006, C. Quesne and V. M. Tkachuk introduced a two-parameter extension of the covariant Heisenberg algebra in a $(D+1)$-dimensional space-time [15]. There are many papers about reformulation of quantum field theory from the viewpoint of the Quesne-Tkachuk algebra. For a review, we refer the reader to Refs. [16,17].

[TABLE]

Where $\mu,\nu=0,1,2,\ldots,D$, $\hat{X}^{\mu}$ and $\hat{P}^{\mu}$ are the generalized position and momentum operators, and $\hat{\textbf{P}}^{2}=\hat{P}_{\mu}\hat{P}^{\mu}=(\hat{P}_{0})^{2}-\sum_{i=1}^{D}(\hat{P}^{i})^{2}$. In the coordinate representation, the generalized position and momentum operators $\hat{X}^{\mu}$ and $\hat{P}^{\mu}$ in Eqs. (6)-(8) have the following exact representations [14]:

[TABLE]

where $\hat{x}^{\mu}$ and $\hat{p}^{\mu}$ are the conventional position and momentum operators which satisfy the conventional covariant Heisenberg algebra $(i.e.,\left[\hat{x}^{\mu},\hat{p}^{\nu}\right]=-i\hbar\eta^{\mu\nu}\text{and}\left[\hat{x}^{\mu},\hat{x}^{\nu}\right]=\left[\hat{p}^{\mu},\hat{p}^{\nu}\right]=0)$. In Eq. (10) $\hat{\textbf{p}}^{2}=-\hbar^{2}\square$. Equations (9) and (10) show that in order to reformulate quantum field theory from the viewpoint of covariant Kempf-Mangano algebra, the conventional position and derivative operators $(\hat{x}^{\mu},\partial_{\mu})$ must be replaced as follows:

[TABLE]

It is important to note that in the limit of $\hbar\ell\rightarrow 0$, the generalized derivative operator $\nabla_{\mu}$ in Eq. (12) becomes the conventional derivative operator, i.e., $\lim_{\hbar\ell\to 0}\nabla_{\mu}=\partial_{\mu}$. In the next section, we will introduce a Lorentz-invariant infinite derivative scalar field theory in the framework of the covariant Kempf-Mangano algebra.

3 Lagrangian Formulation of an Infinite Derivative Scalar Field Theory Based on the Covariant Kempf-Mangano Algebra

The Lagrangian density for a real scalar field in a $(D+1)$-dimensional flat space-time can be written as follows [18]:

[TABLE]

Using Eq. (11) together with the transformation rule for a scalar field, we obtain

[TABLE]

If we use Eqs. (12)-(14), we will get the generalized Lagrangian density for a real scalar field as follows:

[TABLE]

Now, let us consider a classical scalar field theory whose action functional is given by [19]

[TABLE]

The variation of (16) with respect to $\phi$ leads to the following generalized Euler-Lagrange equation [19]

[TABLE]

where

[TABLE]

[TABLE]

If we substitute (15) into (17), we will obtain the following generalized Klein-Gordon wave equation

[TABLE]

Note that in the low-energy limit $(\hbar\ell\rightarrow 0)$, the generalized Klein-Gordon equation (20) becomes the conventional Klein-Gordon equation, i.e.,

[TABLE]

The generalized Klein-Gordon equation (20) has a plane-wave solution as follows:

[TABLE]

where $A$ is the amplitude of the scalar field. After inserting Eq. (22) into Eq. (20), we obtain the following generalized dispersion relation:

[TABLE]

Equation (23) leads to the following generalized energy-momentum relations:

[TABLE]

[TABLE]

where the effective masses $m_{+}(\ell)$ and $m_{-}(\ell)$ have the following definitions

[TABLE]

[TABLE]

In order to obtain the real values for the effective masses $m_{\pm}(\ell)$ in Eqs. (26) and (27) the parameter $\ell$ must satisfy the following inequality

[TABLE]

It must be emphasized that the parameter $\hbar\ell$ which has a dimension of $[length]$ defines a characteristic length scale $\delta:=\hbar\ell$ in our model. Equation (28) shows that the upper limit of $\delta$ is

[TABLE]

If we expand the effective masses (26) and (27) around $\delta=0$, we will obtain the following low-energy expressions for $m_{\pm}(\ell)$

[TABLE]

[TABLE]

Therefore, the low-energy limit of our model describes two particles, one with the usual mass $m$ and the other a ghostlike particle of mass $\frac{2}{mc^{2}\ell^{2}}$.222The appearance of these ghostlike particles in the theory of fourth-order derivative wave equations such as $\square\phi(x)+(\frac{mc}{\hbar})^{2}\phi(x)+\frac{1}{\Lambda^{2}}\square\square\phi(x)=0$ ($\Lambda$ is a regulator (cutoff)) is a well-known problem in higher-derivative quantum field theories [16,20].

4 Relationship between the Low-Energy Behavior of the Model for a Spatially Homogeneous Field Configuration and the Pais-Uhlenbeck Oscillator

In this section, we want to study the low-energy behavior of the infinite derivative scalar field theory which was introduced in the previous section for a spatially homogeneous field configuration. The action functional (16) for the generalized Lagrangian density (15) is

[TABLE]

For a spatially homogeneous real scalar field $\phi(t)$ Eq. (32) becomes

[TABLE]

where $V_{D}=\int_{\mathbb{R}^{D}}d^{D}x$ is the volume of the spatial part of the space-time and dot denotes derivative with respect to $t$.

Using the field redefinition $\psi(t):=\frac{\sqrt{V_{D}}}{c}\phi(t)$ Eq. (33) can be rewritten as follows:

[TABLE]

The action functional $S_{\delta}[\psi]$ in Eq. (34) has the following low-energy expansion

[TABLE]

where

[TABLE]

If we neglect terms of order $\delta^{4}$ and higher in Eq. (35) and dropping out the boundary term $\int_{\mathbb{R}}\ dt\frac{d\Omega(t)}{dt}$, we will find

[TABLE]

Straightforward but tedious calculations show that $S_{\delta}[\psi]$ in Eq. (37) can be written as follows:

[TABLE]

where the effective frequencies $\omega_{\pm}$ are defined as follows:

[TABLE]

[TABLE]

The action functional (38) is a well-known model in the theory of higher-order time derivative models and is called the Pais-Uhlenbeck (PU) oscillator [4,21-25].333The Pais-Uhlenbeck oscillator describes a one-dimensional harmonic oscillator coupled to a higher-order time derivative term whose action functional is

$\displaystyle S_{PU}={\gamma\over 2}\int_{\mathbb{R}}\ dt\ \left[\ddot{\psi}^{2}(t)-(\omega_{+}^{2}+\omega_{-}^{2})\dot{\psi}^{2}(t)+\omega_{+}^{2}\omega_{-}^{2}\psi^{2}(t)\right],$

where $\gamma$ is an arbitrary parameter [21]. This model has a wide applications in several areas of theoretical physics [21-25].

Therefore, in the low-energy limit $(\delta\rightarrow 0)$ our model behaves like a Pais-Uhlenbeck oscillator for a spatially homogeneous field configuration $\phi(t,\vec{\textbf{x}})=\phi(t)$. In order to obtain the real values for the effective frequencies $\omega_{\pm}$ in Eqs. (39) and (40) the characteristic length scale $\delta$ must satisfy the following condition

[TABLE]

According to Eq. (41) the upper limit of $\delta^{PU}$ is

[TABLE]

Equations (29) and (42) show that the upper limit of the characteristic length scale $\delta$ in this work is proportional to $\frac{\hbar}{mc}$, i.e.,

[TABLE]

5 Summary and Conclusions

More than 70 years ago the American physicist H. S. Snyder introduced a one-parameter extension of the covariant Heisenberg algebra in a four-dimensional space-time in order to remove the infinities which appear in quantum field theories [26]. In 2006 a two-parameter extension of the covariant Heisenberg algebra in a $(D+1)$-dimensional Minkowski space-time was presented by Quesne and Tkachuk [15]. The Quesne-Tkachuk algebra contains the Snyder algebra as a subalgebra [15]. In addition, the reformulation of Maxwell equations and Dirac equation from the viewpoint of the Quesne-Tkachuk algebra have been studied for the first time in Ref. [27].

In 2017, G. P. de Brito and his co-workers introduced a modification of the Quesne-Tkachuk algebra [14]. This modified algebra is a covariant generalization of the Kempf-Mangano algebra in a $(D+1)$-dimensional Minkowski space-time. In this work, by using the methods of Ref. [27], after Lagrangian formulation of an infinite derivative scalar field theory in the framework of the covariant Kempf-Mangano algebra, it was shown that the infinite derivative field equation (20) describes two particles with the effective masses $m_{\pm}(\ell)=\frac{1\pm\sqrt{1-2m^{2}c^{2}\ell^{2}}}{mc^{2}\ell^{2}}$. We showed that in the low-energy (large-distance) limit the infinite derivative scalar field theory in Eq. (32) for a spatially homogeneous field configuration $\phi(t,\vec{\textbf{x}})=\phi(t)$ behaves like a Pais-Uhlenbeck oscillator with the effective frequencies $\omega_{\pm}={c\over 2\delta}\left[\sqrt{1+{2mc\delta\over\hbar}}\pm\sqrt{1-{2mc\delta\over\hbar}}\right]$, where $\delta=\hbar\ell$ is the characteristic length scale in this paper. Our calculations in Sections 3 and 4 show that the upper limit of $\delta$ must be proportional to $\frac{\hbar}{mc}$ (see Eq. (43)).

Now, let us evaluate the numerical value of $\delta_{max}$ in Eq. (43).

In nuclear and low-energy particle physics a real scalar field theory describes a neutral $\pi^{0}$ meson [28]. The mass of the $\pi^{0}$ meson is [18]

[TABLE]

Inserting (44) into (43), we find

[TABLE]

It should be noted that the numerical value of $\delta_{max}$ in Eq. (45) is near to the nuclear scale (see page 174 in Ref. [29]), i.e.,

[TABLE]

The above estimations show that in the low-energy limit, the conventional real scalar field theory in Eq. (13) is recovered, while in the high-energy limit the real Klein-Gordon theory in Eq. (13) must be replaced by Eq. (15), i.e.,

[TABLE]

The action functional (32) in the presence of an external current $J(x)$ is

[TABLE]

The action functional (48) can be rewritten as follows:

[TABLE]

where $\nabla_{\mu}$ has been defined in Eq. (12) and $\Upsilon^{\mu}$ has the following definition

[TABLE]

After dropping out the boundary term ${1\over c}\int_{\mathbb{R}^{1,D}}d^{D+1}x\ \partial_{\mu}\Upsilon^{\mu}$ in (49), we will find

[TABLE]

A comparison between the action functionals (2) and (51) shows that for $V\left(\phi(x)\right)={1\over 2}\left(\frac{mc}{\hbar}\right)^{2}\phi^{2}(x)$ and $F(\square_{x})=-\frac{\square_{x}}{(1-\frac{\delta^{2}}{2}\square_{x})^{2}}$ there is an equivalence between a non-local real scalar field theory and an infinite derivative real scalar field theory in the framework of the covariant Kempf-Mangano algebra.444Note that the form factor $F(z)=-\frac{z}{(1-\frac{\delta^{2}}{2}z)^{2}}$ is not an entire function. It must be emphasized that there are many examples in the literatures about non-local quantum field theory in which the form factor $F(z)$ is not an entire function (see Refs. [10,30,31] for more details).

Acknowledgments

We would like to thank the referee for his/her careful reading and constructive comments.