Aharonov-Bohm scattering for relativistic particles in (3 + 1)-dimensional noncommutative space with spin dependence

TL;DR

This paper investigates how noncommutative spacetime with spin effects influences Aharonov-Bohm scattering of relativistic particles, revealing Lorentz-preserving models with unique noncommutativity parameters.

Contribution

It introduces a Lorentz-symmetric noncommutative model incorporating spin in (3+1) dimensions and computes the scattering amplitude for small magnetic flux.

Findings

Noncommutativity affects scattering amplitudes.

Model preserves Lorentz symmetry despite nonconstant noncommutativity.

Comparison with canonical noncommutativity highlights differences.

Abstract

We study the effects of noncommutativity of spacetime with mixed spatial and spin degrees of freedom in a relativistic context. Using the Dirac equation in (3+1) dimensions and in a symmetric gauge, we calculate the invariant amplitude for a small magnetic field flux. The parameter {\theta} that characterizes the noncommutativity here is not constant, and the model preserves Lorentz symmetry. A comparison is made with scattering in the context of canonical noncommutativity.

Figures

Equations

$$[{\hat{x}}_{\mu },{\hat{x}}_{\nu }]=i{\theta }_{\mu \nu },$$

$$\left[{\hat{x}}^{\mu },{\hat{x}}^{\nu }\right]=-i\theta {ϵ}^{\mu \nu \rho \sigma }{S}_{\rho \sigma }+i\frac{{\theta }^2}{2}{ϵ}^{\mu \nu \rho \sigma }{W}_{\rho }{p}_{\sigma },$$

$$W^{\mu }=\frac{1}{2}{ϵ}^{\mu \nu \rho \sigma }{S}_{\rho \sigma }=\frac{1}{2}{\gamma }^5{\sigma }^{\mu \nu }{\partial }_{\nu },$$

$${\gamma }^0=\left(\begin{array}{cc}\mathbf{I}& 0\\ 0& -\mathbf{I}\end{array}\right),{\gamma }^i=\left(\begin{array}{cc}0& {\sigma }^i\\ -{\sigma }^i& 0\end{array}\right),$$

$$S^{0i}=\frac{i}{4}[{\gamma }^0,{\gamma }^i],S^{ij}=\frac{i}{4}[{\gamma }^i,{\gamma }^i],$$

$$x^{\mu }$$

$$p^{\mu }$$

$$\left\{-i{\gamma }^{\mu }{\partial }_{\mu }+M+\frac{e}{2}\left[A_{\mu }(x){\gamma }^{\mu }\star +A_{\mu }(x)\star {\gamma }^{\mu }\right]\right\}\psi (x)=0,$$

$$f({\hat{x}}^{\mu })\star \psi =f(x^{\mu }1+\theta {\gamma }^5{\sigma }^{\mu \nu }{\partial }_{\nu })\psi =f(x^{\mu })\exp \left(\theta {\overleftarrow{\partial }}_{\mu }{\gamma }^5{\sigma }^{\mu \nu }{\overrightarrow{\partial }}_{{}_{\nu }}\right)\psi .$$

$$\left[{\alpha }_i\left(p_i-e{A}_i\right)+\beta M\right]\psi (j,x)=i{\partial }_t\psi (j,x),$$

$$e{A}_i=-\frac{e\Phi }{2\pi }\frac{{ϵ}_{ij}{x}_j}{{\rho }^2}=-\alpha \frac{{ϵ}_{ij}{x}_j}{{\rho }^2},$$

$$\left[-i{\gamma }^0{\gamma }^i{\partial }_i+M{\gamma }^0+e{\gamma }^0{\gamma }^i{A}_i(x)+\frac{e\theta }{2}\left({\partial }_k{A}_i\right){\gamma }^0\left\{{\gamma }^i,{\gamma }^5{\sigma }^{kj}\right\}{\partial }_j\right]\psi (x)=i{\partial }_t\psi (x).$$

$$H_0$$

$$H_{\mathrm{int}}$$

$$H_{\mathrm{int}}=$$

$$+\left(\frac{1}{{\rho }^2}-\frac{2x_2^2}{{\rho }^4}\right)\left(\begin{array}{cc}-i{\sigma }^2& 0\\ 0& -i{\sigma }^2\end{array}\right){\partial }_1+\frac{2x_1{x}_2}{{\rho }^4}\left(\begin{array}{cc}i{\sigma }^1& 0\\ 0& i{\sigma }^1\end{array}\right){\partial }_1\}$$

$$H_{\mathrm{int}}(\rho ,\phi )=-\frac{\alpha \theta }{{\rho }^3}\left(\begin{array}{cccc}0& e^{-i\phi }{z}_{ij}^{+}& 0& 0\\ e^{i\phi }{z}_{ij}^{-}& 0& 0& 0\\ 0& 0& 0& e^{-i\phi }{z}_{ij}^{+}\\ 0& 0& e^{i\phi }{z}_{ij}^{-}& 0\end{array}\right),$$

$$z_{ij}^{\pm }=\left(i{ϵ}_{ij}\pm {\delta }_{ij}\right)x_i{\partial }_j.$$

$$\psi (j,x)=\frac{1}{2\pi }\frac{1}{\sqrt{2E_p}}{e}^{-i{E}_{p}t+i{p}_{3}z}{e}^{im\phi }{e}^{i\pi |\ell |/2}\left(\begin{array}{c}\chi \\ \zeta \end{array}\right),$$

$$\chi$$

$$\zeta$$

$$(p^{\perp }{)}^2=p^2-p_3^2,s=\pm \sqrt{1+\frac{p_3^2}{M^2}}.$$

$$S_{fi}^{(1)}$$

$$=-i\int dt{d}^{3}x{\psi }_f^{†}(x)H_{\mathrm{int}}{\psi }_i(x)e^{i(E_f-E_i)t}{e}^{i(p_3^{\prime }-p_3)z}$$

$$=-i(2\pi {)}^{2}i\delta (E_f-E_i)\delta (p_3^{\prime }-p_3)f_{fi}^{(1)}.$$

$$f_{fi}^{(1)}=\int \rho d\rho d\phi {\psi }_f^{†}(p^{\perp }\rho )H_{\mathrm{int}}{\psi }_i(p^{\perp }\rho ).$$

$$\sqrt{E_p+sM}$$

$$\sqrt{E_p-sM}$$

$${ϵ}_{\ell }$$

$$J_{{\nu }_i}$$

$$S_{fi}^{(1)}=-\frac{ie\theta \Phi }{8\pi E_{p}s}{\int }_0^{\infty }{\int }_0^{2\pi }\frac{1}{{\rho }^2}d\rho d\phi {\mathbf{v}}^{†}\left(\begin{array}{cccc}0& e^{-i\ell \phi }{z}_{ij}^{+}& 0& 0\\ e^{i\ell \phi }{z}_{ij}^{-}& 0& 0& 0\\ 0& 0& 0& e^{-i\ell \phi }{z}_{ij}^{+}\\ 0& 0& e^{i\ell \phi }{z}_{ij}^{-}& 0\end{array}\right)\mathbf{u},$$

$$\mathbf{u}$$

$$\mathbf{v}$$

$$S_{fi}^{(1)}=\frac{e\theta \Phi }{4\pi E_{p}s}\mathcal{W}{\int }_0^{\infty }{\int }_0^{2\pi }d\rho d\phi$$

$$-{ϵ}_{\tilde{\ell }}\frac{1}{{\rho }^2}[\ell J_{{\tilde{\nu }}_2}{J}_{{\nu }_1}-J_{{\tilde{\nu }}_2}{J}_{{\nu }_1}^{\prime }{p}^{\perp }\rho ]e^{-i(\tilde{\ell }+1)\tilde{\phi }+i(\ell +1)\phi }\},$$

Full text

I Introduction

In recent years, several aspects of noncommutative quantum theories have been investigated (see Refs. RevModPhys.73.977, for a review). In these studies, noncommutativity is considered to be a fundamental description of spacetime at the Planck scale CommMathPhys.172.187 or as a particular limit of a quantum theory of gravity.JHEP.09.032

Noncommutative coordinates were first proposed as long ago as 1947 in an attempt to remove the ultraviolet divergences that plague perturbative quantum field theories,PhysRev.71.38 although related ideas were discussed even earlier than that. NuclPhys.B108.30 However, the idea was abandoned after the great success of the renormalization program.

With the advent of string theory, new interest in noncommutative (NC) space emerged. PhysRev.D84.045002 ; PhysRev.D86.105035 ; JPhys.A.46.245303 ; ModPhysLett.A.29.1450048 ; EurPhysJ.C74.3101 ; PhysLett.A378.3509 ; PhysLett.B743.184 ; PhysLett.B761.462 ; JHEP.01.049 ; PhysLett.B766.181 ; PhysLett.A382.3317 ; EurPhysJ.C79.300 ; UkrJPhys.64.131 It has been shown that the dynamics of open strings in the presence of an antisymmetric $B$-field can be described in certain limits by a gauge theory on a noncommutative spacetime, JHEP.09.032 namely, one deformed by the Moyal product (whose definition will be given shortly).

One well-known physical example of noncommutativity of coordinates appears in quantum mechanics. The Landau problem PhysLett.B718.1475 in which a charged particle moves in a strong constant magnetic field that is perpendicular to the plane of motion leads to a noncommutative space when it is projected to a lower Landau level.

At distances of the order of the Planck length, owing to the generation of intense gravitational fields, the measurement of coordinates loses all meaning, as does the concept of “point.” This suggests the introduction of position operators that do not commute:

[TABLE]

where $\theta_{\mu\nu}$ is a constant antisymmetric matrix of dimension (length)2 and the hatted quantities are position operators in NC space. The nonlocality inherent to Eq. (1) leads to the appearance of several significant effects, such as ultraviolet/infrared (UV/IR) mixing. JHEP.03.035 In this context, another interesting aspect of NC spacetime is the violation of unitarity and causality when $\theta_{0i}\neq 0$. NuclPhys.B591.265 ; PhysLett.B533.178 ; EurPhysJ.C25.469 ; JHEP10.2004.072 ; JHEP.06.044 ; JHEP.05.040 ; PhysRev.D69.065008 However, in all these situations, $\theta_{\mu\nu}$ is constant, i.e, we have canonical noncommutativity. It is therefore appropriate to consider novel cases where, for example, the noncommutativity is spin-dependent PhysLett.B680.384 . We should emphasize here that the NC Aharonov–Bohm (AB) effect has been studied in both the quantum mechanical PhysRev.D76.085008 and field theory PhysRev.D70.085005 ; PhysRev.D71.107701 contexts. More general situations may also be considered where $\theta_{\mu\nu}$ is position-dependent, i.e., a new operator. In this sense, Eq. (1) may be taken as a first approximation to a more general setting where the commutator of coordinates itself is not a constant operator. AnnPhys.260.224 ; PhysLett.B.555.83 ; EurPhysJ.54.325 ; PhysRev.D79.125011 In Refs. PhysLett.B718.1475, and JPhys.A.MathTheor.43.285301, , the physical meaning of spin noncommutativity is discussed in terms of a nonstandard Heisenberg algebra.

In the present work, we shall consider a relativistic quantum mechanical model where the spin dependence is incorporated into the Pauli–Lubanski vector. We calculate the scattering amplitude for relativistic particles in a symmetric gauge. The aim of this work is to study this novel kind of noncommutativity and its physical implications in contrast to the canonical case.

II The Aharonov–Bohm effect: symmetric gauge and spin noncommutativity

We consider the AB effect, with an infinitely thin solenoid lying along the $z$ axis in an NC space and with coordinates $\hat{x}^{\mu}$ satisfying the algebra

[TABLE]

where

[TABLE]

is the Pauli–Lubanski pseudovector, $S_{\rho\sigma}=\sigma_{\rho\sigma}/2$ is the spin operator, and $\sigma^{\mu\nu}=\frac{i}{2}[\gamma^{\mu},\gamma^{\nu}]$. Whenever necessary, we assume the Pauli representation for the gamma matrices:

[TABLE]

where the spacetime metric is $\eta^{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$ and $\sigma^{i}$ are the Pauli matrices. The boost and rotation generators are AnIntroductiontoQuantumFieldTheory.41

[TABLE]

so the antisymmetric $S^{ij}=\sigma^{ij}/2$ and $\gamma^{5}=i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}$. The algebra in Eq. (2) can be implemented through the following definitions:

[TABLE]

where $x^{\mu}$ and $p^{\mu}$ satisfy the usual Heisenberg algebra.

We write the NC Dirac equation in a form that ensures a Hermitian Lagrangian density:PhysLett.B718.1475

[TABLE]

where the “star operation” (Moyal product) represents the following action of a Weyl-ordered operator $f(\hat{x}^{\mu})$ on a spinor $\psi(x)$:

[TABLE]

We adopt the symmetric ordering shown in Eq. (7) because the operator $A_{\mu}(\hat{x}^{\mu})$ has a matrix structure that does not commute with $\gamma^{\mu}$. In the relativistic case, the NC Dirac equation describes a spin-1/2 particle in an external electromagnetic field in the presence of spin noncommutativity.

In ordinary relativistic quantum mechanics, the Dirac equation for a particle of mass $M$ and charge $-e$ in an external magnetic field reads

[TABLE]

where $\alpha_{i}=\gamma^{0}\gamma^{i}$ and $\beta=\gamma^{0}$. The magnetic field is derived from a vector potential $(A^{0}=A^{3}=0)$ such that

[TABLE]

where $\Phi$ is the magnetic flux, $\epsilon_{ij}$ is the Levi-Civita symbol (normalized as $\epsilon_{12}=1$), and $\rho=\sqrt{x_{1}^{2}+x_{2}^{2}}$ . This is the gauge potential (symmetric gauge) in an ordinary commutative space, and it corresponds to $\vec{B}=\Phi\delta(\rho)\hat{k}$.

Thus, after performing the star product in the NC Dirac equation in the AB potential, up to first order in $\theta$, Eq. (7) reads

[TABLE]

We are using natural units, such that $\hbar=c=1$, and $e<0$ for the electron charge. A straightforward calculation in the symmetric gauge (10) leads to the total Hamiltonian $H$ as the sum of

[TABLE]

After all the $\gamma$ products have been carried out and the $x_{i}$ and $\partial_{i}$ terms have been properly grouped, $H_{\mathrm{int}}$ reduces to

[TABLE]

(for details, see Appendices B and C). After some algebra, this interaction Hamiltonian may be written as

[TABLE]

where

[TABLE]

The Dirac spinor solution of the Dirac equation in an AB potential, Eq. (9), may be written as PhysRev.D53.2178

[TABLE]

where $m$ is the integer part of the parameter $\alpha=e\Phi/2\pi$ in Eq. (10): $\alpha=m+\delta$, with $0<\delta<1$. The two-component eigenvectors are given by

[TABLE]

where $J_{|\ell-\delta|}$ is the Bessel function of the first kind, $\epsilon_{3}:=\mathrm{sign}(sp_{3})$, $\ell$ is an integer, and $s$ is the eigenvalue of the equation $\hat{S}\psi=s\psi$. In what follows, we take $\mathrm{sign}(sp_{3})=1$. Since $p^{2}=E_{p}^{2}-M^{2}$, we have

[TABLE]

The first-order Born amplitude for the scattering of a Dirac particle induced by spin noncommutativity is then

[TABLE]

This result emerges from the $t$ and $z$ integrations corresponding to energy and momentum conservation in the $z$ direction. Thus, we are left with calculation of the following integral:

[TABLE]

This integration is over all space, and $\psi$ is the Dirac spinor given by Eq. (17). Furthermore, we introduce the following notation for some terms in the spinor components:

[TABLE]

[TABLE]

and similarly for their Hermitian adjoints $\epsilon_{\tilde{\ell}}$ and $J_{\tilde{\nu}_{i}}$ ($\tilde{\ell}$ is the azimuthal quantum number of the outgoing wave). With these considerations, the invariant amplitude can be written as

[TABLE]

where

[TABLE]

and $\dagger$ indicates the Hermitian conjugate. Here, $\tilde{\varphi}=\varphi-\Omega$, where $\Omega$ is the scattering angle (see Fig. 1).

Equation (27) involves spatial derivatives given by the $z$ operators defined in Eq. (16). After multiplying all the matrices, we get

[TABLE]

where $\mathcal{W}=\surd{E_{p+}}\surd{E_{p-}}\surd{s_{+}}\surd{s_{-}}$. The above expression can be cast as

[TABLE]

where the prime denotes differentiation with respect to the argument of the Bessel functions of the first kind, which are given explicitly in Eqs. (18) and (19). We can write $\tilde{\varphi}$ in terms of the scattering angle $\Omega$, and in this way we can easily perform the $\varphi$ integration, yielding

[TABLE]

where we have introduced a parameter $\lambda$ so that, after all the integrations have been performed, we can investigate whether its physical value $\lambda=1$ can be taken. The $\rho$ integrals can be performed GradshteynRyzhik using the analytical continuation of the formula

[TABLE]

where the arguments in the gamma functions and the calculation method are shown explicitly in Appendix A.

After performing all the integrations in Eq. (II), we obtain for the invariant amplitude

[TABLE]

After some manipulations, we find from Eq. (20) that

[TABLE]

where $p^{\perp}=|\vec{p}^{\perp}|$ and the $z$ component of momentum are related to the relativistic energy by $E_{p}^{2}=(p^{\perp})^{2}+p_{3}^{2}+M^{2}$.

III Conclusions

In this paper, we have studied spin noncommutativity for a relativistic system through deformation of the spacetime coordinates $x^{\mu}$ into a nonstandard Heisenberg algebra given by Eq. (2).

In the nonrelativistic situation, the AB scattering amplitude for spin-$1/2$ particles subject to this kind of noncommutativity was calculated in Ref. PhysLett.B680.384, . Generalizing that study to the case of spin-$s$ relativistic particles, we have calculated in $(3+1)$ dimensions the scattering amplitude in the AB symmetric gauge.

For canonical noncommutativity, the AB problem for a solenoid of finite radius was considered in Ref. PhysRev.D66.045018, . In that regime, the gauge potential was expanded in powers of the noncommutativity parameter and the magnetic flux $\Phi$. There, the first order in $\theta$ appeared multiplied by $\Phi^{2}$, and thus, since we are considering here a small magnetic flux, this term is outside the scope of the present model.

We have restricted attention to the potential as written in Eq. (10). In our calculations, because of symmetry considerations, the energy and $p_{3}$ are conserved.

From Eq. (34), we see that the scattering amplitude depends explicitly on $\delta$, the noninteger part of $\alpha$, the term involving the flux. This is quite different from the case of canonical noncommutativity in $(2+1)$ dimensions, where $\delta$ is absorbed into $\alpha$. PhysRev.D76.085008

The canonical NC algebra in Eq. (1) has generated many questions regarding violations of unitarity and causality. This is primarily a consequence of the NC parameter $\theta$ being a constant matrix. However, the present model considering spin noncommutativity has the advantage over the canonical one in that it conserves Lorentz symmetry, since the algebra in Eq. (2) is written in a covariant way. In addition, there is no UV/IR mixing; i.e., the limit $\theta\rightarrow 0$ is smooth.

The study of relativistic or even nonrelativistic quantum mechanics with NC coordinates allows the examination of phenomena that occur at very small scales and the exploration of their physical consequences in a simple setting. However, it should be noted that physical effects related to quantum gravity will appear only in very high-energy processes.

Acknowledgements.

I would like to thank Professor M. Gomes for useful discussions and suggestions.

Appendix A Integrals of Bessel functions in Eq. (II)

The wave contributions for $\ell\leq 0$ and $\ell>0$, given by the integrals of the double products of Bessel functions in Eq. (II), can be calculated using the following result:

[TABLE]

where

[TABLE]

is formally valid provided that the inequalities

[TABLE]

are satisfied.

A.1 The first integral in Eq. (II)

The contribution for $\ell\leq 0$ leads to the following integral:

[TABLE]

where $\sigma=2\lambda$ and

[TABLE]

The condition (41) (with $\lambda\rightarrow 1$) implies

[TABLE]

i.e.,

[TABLE]

so that for $\ell<0$ (and excluding the case $\ell=0$), we have $0<\alpha<1$. This kind of divergence can be eliminated by not taking $\lambda\rightarrow 1$ directly. Instead, we first perform the integral and then take the limit. The integral in Eq. (42) then becomes

[TABLE]

where $\Gamma(\alpha-\ell+1)=(\alpha-\ell)(\alpha-\ell-1)\Gamma(\alpha-\ell-1)$. The other integrals can be calculated following the same approach.

Appendix B The $z_{ij}$ operator in Eq. (16)

To obtain the expression for the $z_{ij}$ operator in Eq. (16), we have to use expressions for the Pauli matrices $\sigma^{i}$ and remember that the $x_{i}$ ($i=1,2$) are in cylindrical coordinates. We have

[TABLE]

where $\rho^{2}=x_{1}^{2}+x_{2}^{2}.$ The nonzero matrix elements are

[TABLE]

where $z_{ij}^{(\pm)}=(i\epsilon_{ij}\pm\delta_{ij})x_{i}\partial_{j}$.

Appendix C Action of the $z_{ij}^{(\pm)}$ operator on a spinor

It is interesting now to see how the $z_{ij}^{(\pm)}$ operator acts on a function that depends on $\varphi$ and $\rho$. In the context of this work, the following are some appropriate cases:

[TABLE]

The product of the two last matrices in Eq. (27) yields the following nonzero matrix elements:

[TABLE]

where $\epsilon_{ij}\epsilon_{jm}x_{i}x_{j}=\rho^{2}$ and $\epsilon_{jm}\delta_{ij}x_{i}x_{m}=\epsilon_{ij}x_{i}x_{j}=0$.