Klein-Gordon theory in noncommutative phase space

Let us consider the noncommutative algebra arising in the Planck scale for avoiding singularity of spacetime. We introduce the 4-vectors of spacetime and momentum operators are defined by

where the operators with hat denote operators in the noncommutative phase space. Thus, the 4D noncommutative relations are defined

and

where the matrices of the commutative relations (2), (3) and (4) describe the features of the noncommutative phase space. We will use a set of parameters to characterize the noncommutative strength and there exist some constraints of the parameters when we map the noncommutative algebra to the Heisenberg algebra in the following section. We will endow the parameters with physical meanings in the following sections.

The noncommutative relations of the position and momentum operators in (2), (3) and (4) give noncommutative operator algebra beyond the canonical (or Heisenberg) algebra in the canonical quantum mechanics. In general, we have two approaches to implement this algebra. One is to play directly the noncommutative algebra in the noncommutative phase space or based on the Moryal product technique,Bellucci ; Gamboa ; Ho ; Gouba ; Chaichiian1 but it is not easy to reveal the noncommutative effects beyond the canonical quantum mechanics. The other approach is to map this noncommutative algebra to the Heisenberg algebra based on the Seiberg-Witten (SW) map with the Bopp shift technique.Seiberg ; Kokado ; Bastos Here we adopt the Bopp shift to connect the noncommutative algebra to the Heisenberg (or canonical) commutative algebra. Liang1 ; Harko ; Liang2 Thus, the noncommutative relations (2), (3) and (4) can be expressed in terms of the Heisenberg commutative relations with additional effects from the noncommutative phase space.Liang1 ; Harko ; Liang2

Let us define $\widehat{x}^{\mu},\widehat{p}_{\mu}\in\widehat{\mathcal{O}}_{nc}$ living in the noncommutative phase space, where $\widehat{x}^{\mu}:=\left(c\widehat{t},\widehat{\mathbf{x}}\right)\equiv\left(c\widehat{t},\widehat{x},\widehat{y},\widehat{z}\right)$, and $\widehat{p}_{\mu}:=\left(\widehat{p}_{0},\widehat{\mathbf{p}}\right)\equiv\left(\widehat{p}_{0},\widehat{p}_{x},\widehat{p}_{y},\widehat{p}_{z}\right)$, and define $x^{\mu},p_{\mu}\in\mathcal{O}_{H}$ living in the Heisenberg phase space, where $x_{\mu}:=\left(ct,\mathbf{x}\right)\equiv\left(ct,x,y,z\right)$, and $p_{\mu}:=\left(p_{0},\mathbf{p}\right)\equiv\left(ct,p_{x},p_{y},p_{z}\right)$.

The Seiberg-Witten (SW) map with the Bopp shift is defined by

where $\widehat{O}\left(\widehat{x}^{\mu},\widehat{p}_{\mu}\right)$ means any operator expressed in terms of $\left(\widehat{x}^{\mu},\widehat{p}_{\mu}\right)$ in the noncommutative phase space, such as momentum and Hamiltonian operators. They obey the noncommutative relations in (2), (3) and (4), while $O\left(x^{\mu},p_{\mu}\right)$ represents any operator expressed in terms of $\left(x^{\mu},p_{\mu}\right)$. They obey the Heisenberg commutative relations,

Thus, we can obtain a claim in the following.

Claim I: The Bopp shift in (5) is constructed by

such that the noncommutative algebra of operators in (2), (3) and (4) in the noncommutative phase space can be implemented equivalently by the Heisenberg commutative algebra (6) in the canonical phase space.

Proof: The Bopp shift (7) can be rewritten as a matrix form

where

and

Using the Heosenberg commutative relations in (6), the noncommutative relations between the position operators are expressed as

where $\lambda^{\mu\nu}$ is the transposition of $\lambda^{\nu\mu}$. Playing the matrix subtraction in (10), the noncommutative relation is obtained

where

This is the noncommutative relation (2).

Similarly, the noncommutative relations between the momentum operators are expressed as

By the matrix subtraction of $\pi$, the noncommutative relations are obtained

where

This is the noncommutative relation (3).

In the same way, the noncommutative relations between the position and momentum operators are expressed as

Making the matrix product of $\lambda$ and $\pi$, the noncommutative relation can be obtained

where

This representation of the position and momentum operators can be regarded as a Heisenbereg representation of the noncommutative quantum mechanics. The Bopp shift provides an efficient way to do the Heisenberg algebra in the noncommutative phase space even though the Bopp shift is not unitary and canonical. It can be verified that the commutative relations (12), (18) and (15) satisfy the Lorentz invariants.

What are the physical meanings of the noncommutative parameters? In the canonical quantum mechanics, the Planck constant plays a role to quantize the intrinsic physical space of elementary particles, such that the quantum state is described by the Hilbert space. To explore physics in the noncommutative phase space, we endow the noncommutative parameters with physical meaning. Based on the Bopp shift, there are only two independent parameters, $\theta$ and $\eta$. In general, physics involved the noncommutative relations happen in the Planck and universe scales, such as intrinsic spacetime singularities, dark matter and dark energy in universe. In particular, the interaction between photon and gravity implies that there exists a minimum length of spacetime in the Planck scale. The dark energy in cosmology can be regarded as an intrinsic minimum curvature of spacetime, which can be interpreted phenomenally as cosmological constant. Thus, we propose a parameterization scheme of the noncommutative parameters associated with a set of physical constants,

where $\ell_{P}=\sqrt{\frac{\hbar G}{c^{3}}}$ is the Planck length, and $\hbar$ is the Planck constant. $G$ is the gravitational constant. $\Lambda\simeq 10^{-56}cm^{-2}$ is the cosmological constant. Based on this parameter setting, we obtain the matrices of the noncommutative relations,

It should be remarked that in general, how to endow the parameters with physics meanings depends on what problem we solve. The parameter setting (20) is based on physics we concern in Planck and universe scales. In principle, we can adopt other settings when we concern microscopic or macroscopic physics such as the de Broglie wave length, which depends on the particle momentum or energy $\lambda_{d}=\frac{\hbar}{p}$. Thus, the de Broglie wave length as a variable extend the formulation to microscopic regions. In fact, the energy-dependent noncommutative quantum mechanics has be proposed for some applications.Harko

Suppose that the energy-momentum relation in the noncommutative phase space still hold, the Klein-Gordon equation is given by

where the momentum operators obey the noncommutative relations (3) and $\phi\equiv\phi(x^{\mu})$. Hereafter we omit the variables $x^{\mu}$ for convenience. To explore what novel properties coming from the noncommutative phase space, we transform the KG equation to the Heisenberg’s representation based on the SW map (5). Note that the analog of electrodynamic field with the minimum coupling, the momenta in the noncommutative phase space can be rewritten as

where $A_{\mu}$ can be viewed as an effective gauge potential induced by the noncommutative phase space,

Note that $\widehat{p}^{\mu}=g^{\mu\nu}\widehat{p}_{\nu}$, where $g^{\mu\nu}$ is the Lorentz metric. The signature here is $(+,-,-,-)$. Thus, the Klein-Gordon (KG) equation can be rewritten as

where

is the 4-vector momentum in the Heisenberg’s canonical representation. In other words, the operators $x_{\mu}$ and $p_{\mu}$ play the Heisenberg’s commutative relations. The gauge potential $A_{\mu}$ descries some novel effects from the noncommutative phase space.

is the 4-vector potential (or gauge potential) generated from the noncommutative phase space, in which

The gauge transformation is defined by

More explicitly, the gauge transformation is expressed as

It can be verified that the KG equation (27) is gauge invariants and Lorentz covariants. Interestingly, the gauge field coupled with the KG equation does not contain a coupled constant. In other words, the coupled constant is dimensionless.

By expending the product in the KG equation (27) and using (26) and (28), $\partial_{\mu}A^{\mu}=0$ for $\partial_{t}x^{i}=0$. The KG equation can be rewritten as a tensorial form,

where $\partial_{\mu}\partial^{\mu}:=\frac{1}{c^{2}}\partial_{t}^{2}-\nabla^{2}$ is the d’Alembert operator. Since all terms in the KG equation are Lorentz scaler, the KG equation in the noncommutative phase space is Lorentz covariant.

As an analog with electrodynamics, we define the gauge field generated from the gauge potential in the noncommutative phase space,

For $\partial_{t}x^{i}=0$, using the parameterization scheme (20) and plugging the gauge potential (26) into (34), the gauge field is obtained

Interestingly, the effective gauge field depends on the Planck constant and cosmological constant, which is consistent with the interpretation of the dark energy emerged from the cosmological constant. There does not exist a coupled constant between the effective gauge field and dynamical equation. This is quite different from conventional electromagnetic $U(1)$ gauge field and Yang-Mills $SU(2)$ gauge field. Moreover, there exists a natural gauge $\partial_{\mu}A^{\mu}=0$. These features of the effective gauge field show an intrinsic geometry (curvature) of spacetime coupled with the dynamical equation. In other words, this property can be interpreted as a natural coupling between quantum and gravity.

To give the probability current and continuity equation, following the similar steps of those for the KG equation coupled with electromagnetic field, making the product of the complex conjugated wave function $\phi^{*}$ in the left-hand side of (27), subsequently subtracting its complex conjugate, we can obtain the continuity equation,Armin

where $J^{\mu}=(c\rho,\mathbf{j})$ is the 4-vector probability current density. This tensorial form of the current continuity equation implies that it is Lorentz invariant. The probability and current densities are given respectively by

which are analog with the KG equation coupled with electromagnetic field. The tensorial form of the continuity equation (36) means that the current continuity equation is Lorentz covariant. However, $\rho$ is not positive definite because $\frac{\partial\phi}{\partial t}$ is arbitrary for the second-order differential equation. Hence, $\rho$ cannot be interpreted strictly as the probability density and thus $\mathbf{j}$ also cannot be defined strictly as the density of the probability current. Interestingly both $\rho$ and $\mathbf{j}$ depend on the effective gauge potential induced in the noncommutative phase space. What effect of the gauge potential plays in the noncommutative phase space? We will examine the KG equation by the perturbation theory in the following section.

To reveal more physics of the KG equation, we can convert the canonical form of the KG equation to its Hamiltonian form.Armin Let us define

such that we have

Thus, solving (39) to express $\varphi$ and $\chi$ in terms of $\phi$, we obtain

Using (38b), we have

Note that

and using the canonical form of the KG equation (27), the left-hand side of (41) can be rewritten as

Combining (38) and (43), we have

Adding and subtracting (44), we obtain

By rearranging the terms of $\varphi$ and $\chi$, the equation (45) can be rewritten as

This is the Hamiltonian form of the KG equation. Let us define $\psi^{T}:=\left(\varphi,\chi\right)$, the KG equation (46) can be rewritten as the Schrödinger-like equation,

where the Hamiltonian is given by

where $\tau_{2(3)}$ are the Pauli’s matrices. It should be remarked that the Hamiltonian in (48) is not Hermitian because $(i\tau_{2})^{\dagger}\neq i\tau_{2}$. We will discuss this issue in the next section.

Similarly, as an analog with Schrödinger equation, the probability density is given by

and the probability current density is obtained

However, $\rho$ is also not positive definite such that $\rho$ cannot be endowed strictly as the probability density and thus $\mathbf{j}$ also cannot be defined strictly as the current densities.Armin

It should be remarked that the noncommutative effects coming from the noncommutative algebra play an effective gauge potential as an analog with the electromagnetic gauge potential, which modifies the densities of the probability and probability current, the velocity and force. However, strictly speaking, the single particle picture is not well-defined because the Hamiltonian is not Hermitian. One proposes to redefine the inner product for Hermitization of operators, which is called the generalized inner product. The detail discussion on this issue may be found in the book. Armin

To examine the basic properties of the relativistic particle in the noncommutative phase space, we consider a free particle model to investigate some dynamical behaviors. Based on the Heisenberg equation (or called as Heisenberg picture), we have two approaches to define observables. One is that the observables are defined in the noncommutative phase space. We refer it as to Case I, in which the commutative relations are calculated directly by the noncommutative relations in (2), (3) and (4), and then using the SW map to transform the results to the Heisenberg representation. The other is that the observables are defined in the Heisenberg phase space. We refer it as to Case II, in which the commutative relations are calculated by the Heisenberg relations of the SW map. We will discuss these two cases.

Case I: In the noncommutative phase space, the velocity of particle is defined by $\mathbf{\widehat{v}}=\frac{d\mathbf{\widehat{x}}}{dt}$ in the noncommutative phase space. The Hamiltonian in (48) of free particle is given by

Thus, the velocity of particle is given by the Heisenberg equation

where $\langle\widehat{O}\rangle$ means the expectation value of $\widehat{O}$ in state $\psi$. Using the commutative relations (18),(see the Appendix A), we have

where

Thus, putting the Hamiltonian (51) into (52), the velocity is obtained as

Similarly, the force is defined by $\mathbf{\widehat{f}}=\frac{d\widehat{\mathbf{p}}}{dt}$. In the same way, we have

Using the commutative relations (15), (see the Appendix A), we have

Consequently, the force is obtained as

Case II: The velocity of particle is defined by $\mathbf{v}=\frac{d\mathbf{x}}{dt}$ in the Heisenberg phase space. based on the SW map, the Hamiltonian in (48) of free particle is given by

The velocity of particle is given by

Note that $[\mathbf{x},\mathbf{p}^{2}]=2i\hbar\mathbf{p}$ and $[\mathbf{x},\mathbf{A}]=0$, the velocity is obtained as

The force is defined by $\mathbf{f}=\frac{d\mathbf{p}}{dt}$ in the Heisenberg phase space. Using the Heisenberg equation, we have

Note that $\left[\frac{\partial}{\partial x_{\mu}},A_{\nu}\right]=\frac{\partial A_{\nu}}{\partial x_{\mu}}$ and $\frac{\partial A_{\mu}}{\partial x_{\mu}}=0$, we have

Putting the Hamiltonian (59) into (60), the force is obtained as

Let us compare the results from two different starting points. The velocity in (55) is mapped to the Heisenberg representation,