Foldy-Wouthuysen transformation of the generalized Dirac Hamiltonian in a gravitational-wave...

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Foldy-Wouthuysen transformation of the generalized Dirac Hamiltonianin a gravitational-wave background

James Q. Quach*

Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan(Received 2 September 2015; published 21 October 2015)

Goncalves et al. [Phys. Rev. D 75, 124023 (2007)] derived a nonrelativistic limit of the generalized Dirac

Hamiltonian in the presence of a gravitational wave, using the exact Foldy-Wouthuysen transformation.This gave rise to the intriguing notion that spin precession may occur even in the absence of a magnetic

field. We argue that this effect is not physical as it is the result of a gauge-variant term that was an artifact of

a flawed application of the exact Foldy-Wouthuysen transformation. In this paper we derive the correct

nonrelativistic limit of the generalized Dirac Hamiltonian in the presence of a gravitational wave, using

both the exact and standard Foldy-Wouthuysen transformation. We show that both transformations

consistently produce a Hamiltonian where all terms are gauge invariant. Unfortunately however, we also

show that this means the novel spin-precession effect does not exist.

DOI: 10.1103/PhysRevD.92.084047 PACS numbers: 04.30.-w, 03.65.Pm, 04.62.+v, 04.80.Nn

I. INTRODUCTION

Gravitational waves (GWs) are one of the major pre-

dictions of general relativity that has yet to be directly

observed. Light interferometry, such as that used in LIGO

and VIRGO, currently stands as the most promising means

by which to detect GWs. More recently however, matter-

wave interferometry has been proposed as a more sensitive

way to detect GWs because of the increased gravitational

interaction from the massive particles [1–8]. In light of this,

it is important to understand the quantum interaction of

massive particles with GWs in the nonrelativistic limit.

Goncalves et al. [9] was the first to investigate the Dirac

Hamiltonian in the presence of an electromagnetic (EM)

gauge field and GWs, in the nonrelativistic limit. In their paper, they came to the intriguing conclusion that in the

presence of the GW, the particle’s spin may precess even in

the absence of a magnetic field, which they propose could

be the basis for a new type of GW detector. In this paper we

show that this precession cannot be physical and is the

result of a miscalculation.A systematic scheme by which to write down the

nonrelativistic limit of the Dirac Hamiltonian with relativ-

istic correction terms is provided by the Foldy-Wouthuysen

(FW) transformation [10]. The FW transformation is a

unitary transformation which separates the upper and lower

spinor components. In the FW representation, the

Hamiltonian and all operators are block diagonal (diagonal

in two spinors). It is an extremely useful representation

because the relations between the operators in the FW

representation are similar to those between the respective

classical quantities. Two popular variants of the FW

transformation exists: the chiral or exact FW (EFW)

transformation [11–14] and the standard FW (SFW) trans-

formation [10].The EFW transformation is so-called because under

certain anticommutative conditions, the scheme leads toan exact expression containing only even terms: even termsdo not mix the upper and lower spinor components, odd terms do. The EFW transformation has the advantage that its calculation is relatively simpler than the SFW trans-

formation, however the EFW may give rise to parity-variant terms which are difficult to physically interpret [14–17].

The SFW transformation involves multicommutator expan-sions of a unitary transformation, which eliminates the oddcontributions to a desired level of accuracy. As it is aniterative scheme, its calculation can be considerably more

difficult than the EFW scheme. The SFW transformationhowever consistently yields terms that are amenable tophysical interpretation.

To arrive at their conclusion, Goncalves et al. applied theEFW transformation to the Dirac Hamiltonian to produce a Hamiltonian which contains a gauge-variant term. It is thisgauge-variant term that gives rise to the novel spin-precession effect. We contend that this term is an artifact of a flawed application of the EFW transformation. In thispaper we will derive the correct nonrelativistic limit of theDirac Hamiltonian in the background of an EM gauge fieldand GWs, first using the EFW transformation, then as a further check, using the SFW transformation. We show that

both these methods produce a gauge-invariant Hamiltonian.As a resultwe will show that there is no spin-precession whenthere is no magnetic field, contrary to the conclusions of

Goncalves et al.

In Sec. II we write down the generalized Dirac

Hamiltonian with a GW metric. In Sec. III we briefly

review the work of Goncalves et al. before deriving the

correct nonrelativistic limit of the generalized Dirac

Hamiltonian with GWs under the EFW transformation.*[email protected]

PHYSICAL REVIEW D 92, 084047 (2015)

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http://dx.doi.org/10.1103/PhysRevD.92.084047http://dx.doi.org/10.1103/PhysRevD.92.084047http://dx.doi.org/10.1103/PhysRevD.92.084047http://dx.doi.org/10.1103/PhysRevD.92.084047http://dx.doi.org/10.1103/PhysRevD.92.084047


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Section IV derives the same Hamiltonian under the SFW

transformation, thereby confirming the consistency of the

Hamiltonian. From this Hamiltonian we will also derive the

spin equation of motion.

II. DIRAC HAMILTONIAN WITH

GRAVITATIONAL WAVE METRIC

The wave function ψ of a spin-1=2 particle of rest massm and charge e in an EM and gravitational field obeys thecurved spacetime Dirac equation (SI units),

iℏγ aea μ

∂ μ − Γ μ −

ie

ℏ A μ

ψ ¼ mcψ : ð1Þ

The spacetime metric g μν can be related at every point to a

tangent Minkowski space ηab via tetrads ea μ, g μν ¼ ea μebνηab.

The tetrads obey the orthogonality conditions ea μeνa ¼ δ ν μ,

ea μe μ

b ¼ δ ab. We use the convention that Latin indicesrepresent components in the tetrad frame. The spinorial

affine connection Γ μ ¼ i4 e

aνð∂ μe

νb

þ Γν μσ e

σ b

Þσ ab, whereΓν μσ is the affine connection and σ ab ≡

i2 ½γ a; γ b are the

generators of the Lorentz group. γ a are gamma matrices

defining the Clifford algebra fγ a; γ bg ¼ −2ηab, withspacetime metric signature (−; þ; þ; þ). A μ is the EMfour-vector potential. We use the Einstein summation

convention where repeated indices ( μ; ν; σ ; a ; b ¼f0; 1; 2; 3g) are summed.

The metric for one of the polarization states of the linear

plane GW is

ds2 ¼ −c2dt2 þ dx2 þ ð1 − 2 f Þdy2 þ ð1 þ 2 f Þdz2; ð2Þ

where f ¼ f ðt − xÞ is a function which describes a wavepropagating in the x-direction. Under this metric Eq. (1)can be written in the familiar Schrödinger picture

iℏ∂ tψ ¼ H ψ , where (α ≡ γ 0γ , β ≡ γ 0, p≡ −iℏ∇, andi;…; n ¼ f1; 2; 3g) [9],

H ¼ cα iðpi − eAiÞ − cf α 2ðp2 − eA2Þ þ cf α 3ðp3 − eA3Þþ β mc2: ð3Þ

III. EXACT FOLDY-WOUTHUYSEN

TRANSFORMATION

Central to the EFW transformation is the propertythat when H anticommutes with J ≡ iγ 5 β , fH; J g ¼ 0,under the unitary transformation U ¼ U 2U 1, where(Λ≡ H=

ffiffiffiffiffiffiH 2

p )

U 1 ¼ 1 ffiffiffi2

p ð1 þ J ΛÞ; U 2 ¼ 1 ffiffiffi2

p ð1 þ β J Þ; ð4Þ

the transformed Hamiltonian is even,

UHU þ ¼ 12 β ffiffiffiffiffiffi

H 2p

þ β ffiffiffiffiffiffi

H 2p

β

þ 12


p − β


p β

J

¼n ffiffiffiffiffiffi

H 2p o

even β þ

n ffiffiffiffiffiffiH 2

p oodd

J: ð5Þ

In Eq. (5) we have made use of the fact that the even

and odd components of any operator Q are respectivelygiven by

fQgeven ¼ 1

2ðQ þ β Q β Þ; fQgodd ¼

1

2ðQ − β Q β Þ:

ð6Þ

As β is an even operator and J is an odd operator, Eq. (5)is an even expression which does not mix the positive andnegative energy states. The EFW transformation has the

further benefit that in many cases the odd components of ffiffiffiffiffiffiH 2

p vanish. In practice


p is taken as a perturbative

expansion where the rest mass energy is the dominate term.In Ref. [9], Goncalves et al. report to use the EFW

transformation to arrive at the following nonrelativistic

limit of Eq. (3):

H GOS ¼ 1

2mðδ ij þ 2 fT ijÞ½ðpi − eAiÞðp j − eA jÞ

þ eℏϵ jklσ l∂ kð AiÞ

þ ℏ2m

∂ ið f ÞT jlϵijkσ kðpl − eAlÞ þ mc2; ð7Þ

where T ≡ diagð0;−1; 1Þ and partial derivatives act onlyon the contents of the parentheses which follow. δ ij, ϵijk,

and σ i are the Kronecker delta, Levi-Cività symbol, and

Pauli matrices respectively.

The spin equation of motion is given by iℏd σ i=dt ¼½σ i; H . Calculating the commutation with H GOS and thentaking the ℏ → 0 limit, one arrives at the following semi-

classical spin equation of motion:

d σ idt

¼ em

ϵijkσ kϵ jlm∂ lð Am þ 2 fT mn AnÞ

−

1

mðp j − eA jÞσ k½T kj∂ ið f Þ − T ij∂ kð f Þ: ð8Þ

From Eq. (8), Goncalves et al. claims that even in the

absence of a magnetic field, spin precession may still occur

due to the coupling of the GW to the EM gauge field.

Setting the magnetic field Bi

¼ ϵijk

∂ jð AkÞ ¼ 0 and neglect-ing small ∂ ð f Þ (which is an appropriate approximationfor GWs of astronomical sources on Earth) the spin

precession is

d σ idt

¼ 2em

f ϵijkϵ jlmT mn∂ lð AnÞσ k: ð9Þ

This poses the notion that due to the presence of the

GW, the gauge field can have a physical effect on spin

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precession even in the absence of a magnetic field. We

argue that the contribution to spin precession as presented

by Eqs. (8) and (9) cannot be physical, as it is not

gauge invariant. More generally, the GW correction term

to the magnetic dipole energy in H GOS, i.e. the term

proportional to T ijϵ jklσ l∂ kð AiÞ, is gauge variant under

the usual gauge transformation, Ai → Ai þ ∂ i χ andψ → ei

eℏ χ ψ , where χ is some scalar function. We believe

the gauge-variant term in H GOS is not the result of theEFW transformation, but the result of an erroneous

calculation in the application of the EFW transformation,

as our derivation using the EFW transformation yields no

such gauge-variant term.We begin by writing Eq. (3) in a more convenient

form,

H ¼ β mc2 þ cα jðδ i j þ T i j f Þðpi − eAiÞ: ð10Þ

Hence,

H 2

¼ m2

c4

þ c2

α jα kðδ i j þ T i j f Þ× ½ðδ lk þ T lk f Þðpi − eAiÞðpl − eAlÞ− T lkpið f Þðpl − eAlÞ: ð11Þ

Neglecting the small f 2 order terms, and using the

identity α iα j ¼ iϵijkσ kI2 þ δ ijI4,

H 2 ¼ m2c4 þ c2ðδ ij þ 2 fT ijÞ½ðpi − eAiÞðp j − eA jÞ

þ eℏc2

2 ðδ ij þ 2 fT ijÞϵ jklΣl½∂ kð AiÞ − ∂ ið AkÞ

þℏc2∂ i

ð f

ÞT jlϵijkΣ

k

ðpl − eAl

Þ;

ð12

Þwhere Σk ≡ σ kI2. As the rest mass energy is the dominate

energy term in the nonrelativistic limit the perturbative

expansion of ffiffiffiffiffiffi

H 2p

to O½1=m2 accuracy yields

H EFW ¼ 1

2mðδ ij þ 2 fT ijÞ½ðpi − eAiÞðp j − eA jÞ

þ eℏ4m

ðδ ij þ 2 fT ijÞϵ jklσ l½∂ kð AiÞ − ∂ ið AkÞ

þ ℏ2m

∂ ið f ÞT jlϵijkσ kðpl − eAlÞ þ mc2: ð13Þ

The first and second terms of H EFW involve the kineticand magnetic dipole energies and their corrections due to

the GW. The third term can be thought of as being the GW

analogue of the Schwarzschild gravitational spin-orbit

energy [18]. The last term is the rest mass energy. In

comparison to the H GOS, H EFW has an extra term propor-

tional to ∂ ið AkÞ in the second line of Eq. (13). The presenceof this term has the effect of ensuring gauge invariance in

the Hamiltonian.

One notes that the EFW transformation has been known

to produce spurious parity-violating terms [14–17].Reference [14] argues that these parity-violating terms

are the result of chiral transformation of the U 2 operator,which alters the symmetry properties of the Hamiltonian.

We point out that the gauge-variant term in H GOS is not of this type.

To further our claim that H EFW is the correct

nonrelativistic limit of the Dirac Hamiltonian in a GWbackground, we provide a consistency check by arriving

at the same Hamiltonian using the alternative SFWtransformation [19].

IV. STANDARD FOLDY-WOUTHUYSEN

TRANSFORMATION

The odd and even components of H are respectivelygiven by

O ¼ 12ðH − β H β Þ; E ¼ 1

2ðH þ β H β Þ: ð14Þ

The SFW is a multicommutator expansion of the unitary

transformation U ¼ eiS,

H 0 ¼ H þ i½S; H þ i2

2!½S; ½S; H þ ; ð15Þ

where S ¼ − i β 2m

O. i½S; H ≈ −O generates a term that eliminates the odd operator O, however many moreterms are generated by the higher-order terms which

potentially could be odd operators. To eliminate these

odd operators, the FW transformation is repeated onsubsequent Hamiltonians (i.e. H 0, H 00, H 000, and so on)

until all odd operators are eliminated to the required order of accuracy.For convenience we will work in the natural units where

ℏ ¼ c ¼ e ¼ 1. We will put ℏ, c, e back into the finalequation. We begin by calculating the following commu-

tator relations:

½ βα iðpi − AiÞ; α jðp j − A jÞ¼ 2 βδ ijðpi − AiÞðp j − A jÞ þ 2 βϵijkΣk∂ jð AiÞ; ð16Þ

½ βα iðpi − AiÞ; f α 2ðp2 − A2Þ

¼ 2 β f

ðp2 − A2

Þ2

þ β f ϵi2kΣk

½∂ 2

ð Ai

Þ− ∂ i

ð A2

Þþ β Σ3∂ 1ð f Þðp2 − A2Þ; ð17Þ

½ βα iðpi − AiÞ; f α 3ðp3 − A3Þ¼ 2 β f ðp3 − A3Þ2 þ β f ϵi3kΣk½∂ 3ð AiÞ − ∂ ið A3Þ

þ β Σ2∂ 1ð f Þðp3 − A3Þ: ð18Þ

Using these commutator relations and Eq. (3) we write

down

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i½S; H ¼ β m

δ ijðpi − AiÞðp j − A jÞ þ β

mϵijkΣk∂ jð AiÞ

þ 2 β m

fT ijðpi − AiÞðp j − A jÞ

þ β m

fT ijϵ jklΣl½∂ kð AiÞ − ∂ ið AkÞ

þ

β

m

∂ i

ð f

ÞT jlϵijkΣ

k

ðpl − Al

Þ− α i

ðpi − Ai

Þþ f ½α 2ðp2 − A2Þ − α 3ðp3 − A3Þ þ h:o:; ð19Þ

where h.o. indicates that there are higher order terms.Using Eq. (19),

i½S; H þ i2

2 ½S; ½S; H ¼ 1

2m½ βα iðpi − AiÞ;

1

2α jðp j − A jÞ − f α 2ðp2 − A2Þ þ f α 3ðp3 − A3Þ þ β m

þ 12m

½− β f α 2ðp2 − A2Þ þ β f α 3ðp3 − A3Þ; β m þ h:o:

ð20ÞNeglecting higher order terms,

HSFW ¼ H þ i½S; H þ i2

2 ½S; ½S; H

¼ β m þ β 2m

ðδ ij þ 2 fT ijÞðpi − AiÞðp j − A jÞ

þ β 4m

ðδ ij þ 2 fT ijÞϵ jklΣl½∂ kð AiÞ − ∂ ið AkÞ

þ β 2m

∂ ið f ÞT jlϵijkΣkðpl − AlÞ: ð21Þ

Explicitly reinstating ℏ, c, e, one retrieves Eq. (13)

from Eq. (21). In other words H EFW ¼ H SFW to O½1=m2accuracy, where HSFW ¼ β H SFW. The EFW and SFWtransformations are not equivalent unitary transformations,and in general can give rise to different Hamiltonians in

the nonrelativistic limits. However in the current case, that both the EFW and SFW transformation yields the same

Hamiltonian, is good verification that Eq. (13) is the correct

nonrelativistic limit of the Dirac Hamiltonian in the

presence of an EM gauge and GW field.Importantly, H EFW is gauge invariant. This has conse-

quences for the physical behavior of a spin-1=2 particleinteracting with GWs. In particular there is a stark differ-

ence in the spin-precession behavior described by H GOSand H EFW. The corresponding semiclassical spin equation

of motion under H EFW is

d σ idt

¼ em

ϵijkσ kϵ jlm½∂ lð AmÞ þ fT mn∂ lð AnÞ − fT mn∂ nð AlÞ

−

1

mðp j − eA jÞσ k½T kj∂ ið f Þ − T ij∂ kð f Þ: ð22Þ

Unlike the spin equation of motion derived from H GOS,Eq. (22) is gauge invariant. If one neglects the small ∂ ð f Þand sets the magnetic field to zero then d σ =dt ¼ 0, andthere is no spin precession, in contrast to Eq. (9) and the

claims of Goncalves et al.

V. CONCLUSION

The spin precession of a spin-1=2 particle in theabsence of a magnetic field but in the presence of a

GW is a false effect of a gauge-variant Hamiltonian.

We consistently derived a gauge-invariant Hamiltonian

with both the EFW and SFW transformations, whichwe contend to be the correct nonrelativistic limit of

the generalized Dirac Hamiltonian in the backgroundof a GW spacetime metric. Using this Hamiltonian

we showed the novel spin-precession effect no longer exists.

ACKNOWLEDGMENTS

The author would like to thank M. Lajkó, C.-H. Su,

A. Martin, and S. Quach for discussions and checking themanuscript. This work was financially supported by the

Japan Society for the Promotion of Science. The author isan International Research Fellow of the Japan Society for

the Promotion of Science.

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FOLDY-WOUTHUYSEN TRANSFORMATION OF THE … PHYSICAL REVIEW D 92, 084047 (2015)

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