Effects of Space-Time Curvature on Spin-1/2 Particle Zitterbewegung

Abstract

This paper investigates the properties of spin-1/2 particle Zitterbewegung in the presence of a general curved space-time background described in terms of Fermi normal co-ordinates, where the spatial part is expressed using general curvilinear co-ordinates. Adopting the approach first introduced by Barut and Bracken for Zitterbewegung in the local rest frame of the particle, it is shown that non-trivial gravitational contributions to the relative position and momentum operators appear due to the coupling of Zitterbewegung frequency terms with the Ricci curvature tensor in the Fermi frame, indicating a formal violation of the weak equivalence principle. Explicit expressions for these contributions are shown for the case of quasi-circular orbital motion of a spin-1/2 particle in a Vaidya background. Formal expressions also appear for the time-derivative of the Pauli-Lubanski vector due to space-time curvature effects coupled to the Zitterbewegung frequency. As well, the choice of curvilinear co-ordinates results in non-inertial contributions in the time evolution of the canonical momentum for the spin-1/2 particle, where Zitterbewegung effects lead to stability considerations for its propagation, based on the Floquet theory of differential equations. Comment: 22 pages, no figures; slight revisions; accepted for publication in Classical and Quantum Gravity

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arXiv:0903.1346v2 [gr-qc] 27 Aug 2009
Effects of Space-Time Curvature on Spin-1/2
Particle Zitterbewegung
Dinesh Singh†§
† Department of Physics, University of Regina, Regina, Saskatchewan, S4S 0A2,
Canada
and Nader Mobed†
E-mail: dinesh.singh@uregina.ca and nader.mobed@uregina.ca
Abstract.
This paper investigates the properties of spin-1/2 particle Zitterbewegung
in the presence of a general curved space-time background described in terms of Fermi
normal co-ordinates, where the spatial part is expressed using general curvilinear
co-ordinates. Adopting the approach first introduced by Barut and Bracken for
Zitterbewegung in the local rest frame of the particle, it is shown that non-trivial
gravitational contributions to the relative position and momentum operators appear
due to the coupling of Zitterbewegung frequency terms with the Ricci curvature tensor
in the Fermi frame, indicating a formal violation of the weak equivalence principle.
Explicit expressions for these contributions are shown for the case of quasi-circular
orbital motion of a spin-1/2 particle in a Vaidya background. Formal expressions also
appear for the time-derivative of the Pauli-Lubanski vector due to space-time curvature
effects coupled to the Zitterbewegung frequency. As well, the choice of curvilinear co-
ordinates results in non-inertial contributions in the time evolution of the canonical
momentum for the spin-1/2 particle, where Zitterbewegung effects lead to stability
considerations for its propagation, based on the Floquet theory of differential equations.
PACS numbers: 03.65.Ta, 04.20.-q, 04.20.Cv
1. Introduction
The concept of Zitterbewegung, or “trembling motion” in German due to the rapid
oscillation of a spin-1/2 particle about its classical worldline, is one that still generates
great interest about foundational issues in quantum mechanics, even after over 70 years
since its initial discovery by Schr¨ odinger [1]. When investigating the free-particle motion
of an electron via the Dirac equation, he showed that its instantaneous velocity vinst.
equals the speed of light c, which clearly disagrees with the fact that a massive particle’s
observed speed will always be vobs.< c. However, Dirac convincingly emphasized [2] that
this apparent contradiction is resolved by noting that vobs.is really an average velocity
determined by measuring the electron’s position over a sufficiently small time interval
∆t. When c∆t ≫ ¯ h/mc, the Compton wavelength for the electron of mass m, it follows
that the practical measurement of vobs.always leads to a preservation of causality, and
§ Author to whom correspondence should be addressed.

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Effects of Space-Time Curvature on Spin-1/2 Particle Zitterbewegung2
without any complications due to Zitterbewegung. Nevertheless, this highly non-intuitive
oscillatory behaviour identified to exist for massive spin-1/2 particles is a conceptually
rich feature of relativistic quantum mechanics that still offers both thought-provoking
challenges and opportunities for deeper exploration [3, 4, 5], especially when applied to
more complicated situations.
While past and recent investigations of Zitterbewegung since Schr¨ odinger’s and
Dirac’s studies have been largely motivated by attempts to understand the intrinsic
structure of the electron [6], there currently exists an intensive practical study of
the concept within the context of condensed-matter theory and experiment. Recent
explorations of this phenomenon involving two-dimensional carbon sheets known as
graphene have been proposed [7, 8], where Zitterbewegung effects are potentially
observable in the non-relativistic limit of the Dirac equation, while a second paper
claims a direct observation for photons in a two-dimensional photonic crystal [9], and
a third paper proposes detection of Zitterbewegung using ultracold neutral atoms [10].
These studies are especially interesting given the largely accepted understanding that
free-particle Zitterbewegung arises only in situations where positive- and negative-energy
solutions to the Dirac equation can interact, allowing for quantum interference effects
to manifest the rapid oscillation terms that would otherwise remain decoupled and
unobservable.
This leads to the following consideration where it concerns interactions in a
gravitational background. Given that external fields can introduce non-trivial effects
that lead to observable consequences for electron Zitterbewegung, and given that
gravitation is best described (for now) by Einstein’s theory of general relativity in
terms of the dynamical curvature of space-time due to external matter, it is reasonable
to envision curved space-time as something like a medium with an energy density
content not unlike that of an electromagnetic background.
fundamental distinction that Einstein’s equivalence principle is an integral part of
general relativity, reflected mathematically as diffeomorphism invariance under co-
ordinate transformations. This is certainly reasonable when considering purely classical
descriptions of matter, where it is meaningful to speak about its localization within a
compact spatial region. However, conceptual difficulties arise when considering quantum
matter in a curved space-time background, where issues of non-locality due to the wave-
particle duality of quantum objects overlap with classical descriptions of space-time
curvature. This is a particularly relevant issue considering that gravitational effects are
manifested locally from observing tidal forces via the geodesic deviation equation, which
requires the existence of a neighbouring geodesic in relation to the reference worldline.
It is very unclear how to disentangle the competing effects of quantum non-locality and
the locality of classical gravitation in a way that results in meaningful statements about
quantum matter effects in a curved space-time background, including Zitterbewegung.
The purpose of this paper is to explore the physical properties of spin-1/2
particle Zitterbewegung in a curved space-time background described locally in terms
of Fermi normal co-ordinates. Its motivation is based on understanding whether spatial
Of course, there is the

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Effects of Space-Time Curvature on Spin-1/2 Particle Zitterbewegung3
fluctuations about the particle’s worldline lead to non-trivial gravitational corrections
that influence its propagation in space-time.
formalism of Fermi normal co-ordinates is to describe the spatial fluctuations in terms
of curvilinear spatial co-ordinates [11, 12, 13]. This paper begins with a brief review
of the existing treatment of Zitterbewegung in a flat space-time background, as given in
Section 2. Because the Fermi normal co-ordinate system is defined locally about the
particle’s worldline, it is important to make use of a local perspective for the occurrence
of Zitterbewegung and explore its physical consequences. It so happens that the approach
introduced by Barut and Bracken [14] is particularly well-suited for this purpose, and so
this perspective is adopted for consideration in a curved space-time background. This
leads to a brief introduction of the covariant Dirac equation in Fermi normal co-ordinates
in Section 3, which includes the curvilinear co-ordinate generalization. A presentation
of the corresponding Dirac Hamiltonian derived from within this framework is given
in Section 4, where contributions due to Zitterbewegung reveal a formal violation of
the weak equivalence principle due to direct coupling of the gravitational field with
the particle’s mass. From this Hamiltonian, it becomes possible to compute the time-
evolution of quantum operators in Section 5 and show how Zitterbewegung formally
appears in the expressions. Particular attention is given to the time-evolution of the
position and momentum operators, and also the covariant spin operator described by
the Pauli-Lubanski vector. A brief conclusion follows in Section 6.
For this paper, geometric units of G = c = 1 are adopted, while the curvature
tensor definitions follow the conventions given by Misner, Thorne, and Wheeler [15],
but with metric signature −2. As well, the flat space-time gamma matrices follow the
conventions of Itzykson and Zuber [16].
A recent innovation in the standard
2. Review of Free-Particle Spin-1/2 Zitterbewegung in Flat Space-Time
2.1. General Formalism
Before introducing Zitterbewegung in curved space-time, it is useful to first review the
original treatment of the problem in flat space-time by Schr¨ odinger [1, 2, 14]. Assuming
space-time co-ordinates in Cartesian form Xˆ µ= (T,X), the Schr¨ odinger equation is
i¯ h(∂/∂T)ψ(X) = H0ψ(X), where
H0= mβ(0)+ α(0)· P(0)
(1)
is the free-particle Dirac Hamiltonian defined in terms of spin-1/2 particle mass m,
canonical momentum P(0)
?
?
while the corresponding position operators Xˆ 
ˆ µ = i¯ h∂/∂Xˆ µ, and 4 × 4 matrices αˆ 
= 2ηˆ αˆβ[16]. As expected, the canonical momentum operators satisfy
(0)= γˆ0γˆ and β(0)= γˆ0
withγˆ α,γˆβ?
P(0)
ˆ µ,P(0)
ˆ ν
?
= 0, (2)
(0)satisfy
?
Xˆ ı
(0),Xˆ 
(0)
?
= 0,
?
Xˆ ı
(0),P(0)
ˆ 
?
= −i¯ hδˆ ıˆ . (3)

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Effects of Space-Time Curvature on Spin-1/2 Particle Zitterbewegung4
Adopting the Heisenberg picture, it follows that the time derivative of some operator
A is
dA(T)
dT
=i
¯ h[H0,A(T)] .(4)
Clearly, dPˆ 
operators are
(0)(T)/dT = dH0/dT = 0, while the instantaneous velocity and acceleration
dXˆ 
(0)(T)
dT
= αˆ 
(0)(T), (5)
d2Xˆ 
(0)(T)
dT2
=
dαˆ 
(0)(T)
dT
= −2i
¯ hηˆ 
(0)(T)H0, (6)
where
ηˆ 
(0)(T) = αˆ 
(0)(T) − Pˆ 
(0)(T)H−1
0 .(7)
Because Pˆ 
(0)(T) and H0are constants of the motion, it is also true that
dηˆ 
(0)(T)
dT
=
d
dT
?
αˆ 
(0)(T) − Pˆ 
¯ hηˆ 
(0)(T)H−1
0
?
=
dαˆ 
(0)(T)
dT
= −2i
(0)(T)H0.(8)
An immediate solution follows for ηˆ 
(0)(T), such that
ηˆ 
(0)(T) = ηˆ 
(0)(0)e−2iH0T/¯ h,(9)
or
αˆ 
(0)(T) = Pˆ 
(0)(0)H−1
0
+ ηˆ 
(0)(0)e−2iH0T/¯ h. (10)
Direct integration of (10) then leads to the free-particle solution for Xˆ 
the time-evolution for the position and canonical momentum operators are
(0)(T), such that
Xˆ 
(0)(T) = Xˆ 
(0)(0) +
+¯ h
2ηˆ 
?
Pˆ 
(0)(0)H−1
0
?
T
(0)(0)H−1
0
?
sin(2H0T/¯ h) − 2i sin2(H0T/¯ h)
?
, (11)
Pˆ 
(0)(T) = Pˆ 
(0)(0). (12)
The first two terms of (11) denote the expected mean position as a function of time T,
while the remaining terms show the rapid oscillation effect superimposed about the spin-
1/2 particle’s worldline due to Zitterbewegung. For future reference, it is straightforward
to show that Zitterbewegung also applies to β(0)(T) and the Pauli spin matrix σˆ 
such that
(0)(T),
β(0)(T)= β0e−2iH0T/¯ h+ mH−1
0
?
2 sin2(H0T/¯ h) + i sin(2H0T/¯ h)
?
, (13)
σˆ 
(0)(T)= σˆ 
0+ i
?
α0× P(0)(0)
?ˆ H−1
0
?
2 sin2(H0T/¯ h) + i sin(2H0T/¯ h)
?
, (14)

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Effects of Space-Time Curvature on Spin-1/2 Particle Zitterbewegung5
where
αˆ 
0≡ αˆ 
(0)(0),β0 ≡ β(0)(0),
σˆ 
0≡ σˆ 
(0)(0). (15)
Before continuing, it is worthwhile to explore the mathematical details of the
operators which describe the Zitterbewegung phenomenon.
formulated description of the problem [17], it is possible to establish that the domain D
corresponding to the position operator Xˆ 
that
Based on a precisely
(0)(0) is invariant under time evolution, such
D
?
Xˆ 
(0)(T)
?
= e−iH0T/¯ hD
(0)(T) remains bounded for all finite T.
?
Xˆ 
(0)(0)
?
= D
?
Xˆ 
(0)(0)
?
. (16)
As such, (16) implies that Xˆ 
the expectation value for the time-evolved position operator is well behaved, in that
the component of (11) exhibiting Zitterbewegung integrates to zero in the expectation
value as T → ∞. Furthermore, it is well-known [17, 18, 19] that the Hilbert space
identified with solutions of the free-particle Dirac equation decouples into invariant
subspaces defined by positive and negative energy states, respectively. This implies
that the expectation values of Xˆ 
energy states never reveal any Zitterbewegung-induced effects. Not only does it explain
why Zitterbewegung is not an observable for a propagating free particle, it suggests the
existence of a breakdown in the single-particle treatment of quantum matter, justifying
the need for the standard quantum field theory description.
In addition,
(0)(T) formed by exclusively positive energy or negative
2.2. The Barut-Bracken Approach to Zitterbewegung
The main expressions above can apply to spin-1/2 particle motion in a general reference
frame with Pˆ 
“microscopic” degrees of freedom for position, momentum, and angular momentum to
explain the electron’s rapid oscillatory behaviour as a separable effect superimposed on
its observable motion [1, 14]. To this end, he proposed the existence of a microscopic
position operator ξˆ (T), according to
(0)(T) ?= 0. Schr¨ odinger’s original view of Zitterbewegung was to search for
Xˆ 
(0)(T) = Xˆ 
(A)(T) + ξˆ (T), (17)
where
Xˆ 
(A)(T) = Xˆ 
(0)(0) +
?
Pˆ 
(0)(0)H−1
0
?
T −i¯ h
2ηˆ 
(0)(0)H−1
0 , (18)
ξˆ (T)=i¯ h
2ηˆ 
(0)(0)H−1
0 e−2iH0T/¯ h,(19)
along with a microscopic momentum associated with ξˆ (T), in the form
ηˆ 
(0)(T)H0+ Pˆ 
(0)(T) =
dXˆ 
(0)(T)
dT
H0. (20)
In re-examining Zitterbewegung with the motive to understand the electron’s
intrinsic structure for addressing self-energy and renormalization issues in QED, Barut
and Bracken [14] saw no difficulty with Schr¨ odinger’s formulation of microscopic