Optical cavity tests of Lorentz invariance for the electron

0.34 MB·English

by  Holger Mueller

Optical cavity tests of Lorentz invariance for the electron
Holger Mu¨ller, Sven Herrmann, Alejandro Saenz, and Achim Peters
Institut fu¨r Physik, Humboldt-Universit¨at zu Berlin, 10117 Berlin, Germany.
∗
Tel. [phone], Fax [phone]
Claus La¨mmerzahl
ZARM (Center for Applied Space Technology and Microgravity),
Universit¨at Bremen, Am Fallturm, 28359 Bremen, Germany
A hypothetical violation of Lorentz invariance in the electrons’ equation of motion (expressed
within the Lorentz-violating extension of the standard model) leads to a change of the geometry
of crystals and thus shifts the resonance frequency of an electromagnetic cavity. This allows ex-
perimental tests of Lorentz invariance of the electron sector of the standard model. The material
dependenceoftheeffectallowstoseparateitfromanadditionalshiftcausedbyLorentzviolationin
4
electrodynamics,andtoplaceindependentlimitsonbotheffects. Frompresentexperiments,upper
0
limits on Lorentz violation in theelectrons’ kinetic energy term are deduced.
0
2
PACSnumbers: 11.30.Cp,03.30.+p,04.80.Cc,03.50.De

n
a
J I. INTRODUCTION surement of the resonance frequency ν =mc/(2L) of an

5 electromagnetic(Fabry-P´erot)cavity,thatisgivenbythe
velocityoflightc alongthe cavityaxis,the cavitylength

1 Special relativity and the principle of Lorentz invari- L, and a constant mode number m. Lorentz violation
v ance describe how the concepts of space and time have causingashift ofc orL connectedto arotationorboost
6 tobeappliedwhendescribingphysicalphenomenainflat of the cavity frame of reference can thus be detected
1
space–time. Improving the accuracy of the experimental through the corresponding shift of the resonance fre-
0
1 verification of these fundamental concepts is of great in- quency. Fromsuchexperiments,upperlimitsonatensor
0 terest,alsobecauseaviolationofLorentzinvarianceisbe (kF)κλµν havebeenfound,thatencodesLorentzviolation
4 a feature of many current models for a quantum theory inthephotonicsectoroftheSME[12,13,15,16,17,18].
0 of gravity, e.g., string theory [1, 2], loop gravity [3, 4], Theseexperimentsaremainlybasedontheshiftofccon-
h/ and non–commutative geometry [5]. Such a violation of nected to non-zero values of (kF)κλµν, as an additional
p Lorentz invariance is described in the general standard changeofLcausedby(k ) andacorrespondingori-
F κλµν
- modelextension(SME)developedbyColladayandKost- entation dependent modification of the Coulomb poten-
p elecky´[6]. Accordingtoit,Lorentzviolatingtermsmight tial is negligible for most cavity materials [19].
e
entertheequationsofmotionofbosonsandfermions. At
h In this work, we treat the effect of Lorentz violation
first sight, the quantum gravity induced corrections and
: within the fermionic sector ofthe standardmodelexten-
v effects are of order E/E 10 28 where E is the en-
i QG ∼ − sion in cavity experiments. A modified kinetic energy
X ergy scale of the experiment (which in ordinary optical
term entering the non-relativistic Schr¨odinger hamilto-
experiments is of the order 1 eV) and E E is
ar ofthe orderofthe Planckenergy. TherefQorGe∼thesPeleanffcekcts nian of the free electron (p2 +2Ej′kpjpk)/(2m) leads to
a change of the geometry of crystals, and thus a change
seem to be far from being observable in laboratory ex-
δ ν of the resonance frequency of a cavity made from
e
periments. However, as it occurs e.g. in scenarios lead- th−iscrystal. Here,p isthe3-momentum,mtheelectron
j
ing to a modification of the Newton potential at small
mass,andE = c c δ adimensionless3 3ma-
distances, some mechanism may apply which effectively j′k − jk− 00 jk ×
trix given by a tensor c of the SME. Thus, the total
µν
leads to much larger effects in the laboratory. It is thus
shift of the resonance frequency δν =δ ν +δ ν ,
e res EM res
interestingtofindexperimentalconfigurationsinthelab- whereδ ν denotes the shiftdue to Lor−entzviolationin
EM
oratorythat canplacestrongupper limits onasmanyof
the electromagnetic sector. Since δ ν depends on the
e
these terms as possible. cavity material, it can be be distingu−ished from Lorentz
Experiments on Lorentz symmetry that study light violationinelectrodynamicsbycomparingcavitiesmade
propagationhavea longandfascinatinghistory,starting fromdifferentmaterials. Experimentsusingsuitablecon-
from the original interferometer experiments of Michel- figurations of cavities can place separate upper limits
son [7] in Potsdam and Michelson and Morley [8] in on the components of cµν and (kF)κλµν. Using data
Cleveland. Modern versions of these experiments [9, 10, availablefrompastexperiments, we deduce approximate
11, 12, 13, 14, 15] replace the interferometer by a mea- bounds on some combinations of components of cµν at
the 10 14 level. From future cavity experiments, Earth
−
andspace-based,thatareprojectedastestsofLorentzvi-
olationin electrodynamics [20, 21, 22], bounds at a level
∗Electronicaddress: [email protected] down to 10−18 can be expected.
2
In Sec. II, we present the non–relativistic hamiltonian me and mp are the electron and the proton mass, γ ,γ
ν 5
of the free electron within the SME. Since violations of and σµν are the conventional Dirac matrices, and Dν is
Lorentz invariance are certainly small, it is sufficient to the covariant derivative. In this work, we deal mostly
worktofirstorderintheLorentz–violatingmodifications with electrons and add a superscript to denote param-
throughout. The crystal adjusts its geometry such as to eters for particles other than the electron, e.g., c is
µν
minimize the totalenergy δh +U ,where δh isthe a parameter for the electron and cp the corresponding
h i elast h i µν
expectation value of the Lorentz–violating part of the parameter for the proton. The tensors entering M have
hamiltonian and U is the elastic energy associated the dimension mass, the others are dimensionless. H
elast µν
with distortion of the crystal. It is calculated in Sec. is antisymmetric; g is antisymmetric in its first two
λµν
III. The resulting geometry change is calculated in Sec. indices. c and d aretraceless. Gauge invarianceand
µν µν
IV. A fairly detailed model for the crystal allows us to renormalizability excludes e ,f , and g , so these are
ν ν λµν
obtain specific results for practical materials, including either zero or suppressed relative to the others [23].
quartz and sapphire. In Sec. V, we discuss experimental Lorentzviolationforthephotonsisencodedintheten-
configurations and obtain bounds on Lorentz violation sors (k )κ and (k ) . The four degrees of freedom
AF F κλµν
in the electrons’ equation of motion from present exper- containedin(k )κ areconstrainedstronglyinmeasure-
AF
iments. In appendix A, we discuss the hypothetical case ments of cosmological birefringence [17, 18] and are ne-
of a cavity made from a spin-polarized solid, which al- glected in what follows. 10 of the 19 degrees of freedom
lows to place experimental limits on an additional, spin of(k ) areconstrainedbyastrophysicalobservations
F κλµν
dependent term from the SME, at least in principle. In [17, 18], the other 9 can be measured in cavity experi-
appendix B, we summarize some conventions made in ments [13, 15, 16, 17, 18].
elasticitytheorythatareneededforourcalculations,and
in appendix C, we give in detail the Fourier components
of the signal for Lorentz violation in laboratory experi- B. Modified non-relativistic hamiltonian
ments on Earth.
1. Free electron
II. STANDARD MODEL EXTENSION
The non-relativistic Schr¨odinger hamiltonian h = hˆ +
δh of a single free electron within the SME derived from
A. Model
thisLagrangian(usingFoldy-Wouthuysenmethods,[24])
isthe sumofthe usualfree-particleHamiltonianˆhanda
The SME starts from a Lagrangian formulation of
Lorentz-violating term [23, 24]
the standardmodel, addingallpossibleobserverLorentz
scalarsthatcanbe formedfromthe knownparticles and δh = mA +mB σj +C p +D p σk
Lorentz tensors. Taken from the full SME that contains ′ j′ j′ j j′k j
p p p p
all known particles, the Lagrangian involving the Dirac +E j k +F j kσl (3)
j′k m j′kl m
fields ψe of the electron and ψp of the proton and the
electromagnetic field Fµν can be written as (in this sec- with the components of the 3-momentum p and of the
tion, we use units with ~=c=1; the greek indices take Pauli matrices σj. The latin indices takej the values
the values 0,1,2,3) [6, 23] 1,2,3. (We denote both 3-vectors such as x and recip-
j
rocal 3-vectors such as p by subscript.) A Hamiltonian
i 1 i j
= ψ¯eΓeDνψe ψ¯eMeψe+ ψ¯pΓpDνψp of this form has also been derived in [25]. The constant
L 2 ν − 2 2 ν
term mA has no physical consequences and is included
′
1 1
ψ¯pMpψp+h.c FµνFµν (1) for completeness. The term proportional to Cj′ can be
−2 − 4 eliminatedbychoosingcoordinatessuchthatthesystems
1 1
(k ) FκλFµν + (k )κǫ AλFµν, centre of mass is at rest [25]. The dimensionless coeffi-
F κλµν AF κλµν
−4 2 cients A,B ,C ,D ,E and F can be expressed in
′ j′ j′ j′k j′k j′kl
terms of the quantities entering the Lagrangian[23, 24]:
where h.c. denotes the hermitian conjugate of the previ-
ous terms, and Aλ is the vector potential. The symbols
1
Γe,p and Me,p are given by A = a c e , (4)
ν ′ m 0− 00− 0
Γν = γν +cµνγµ+dµνγ5γµ+eν +ifνγ5+ 12gλµνσλµ, Bj′ = −mbj +dj0− 21εjklgkl0+ 21mεjklHkl, (5)
a
M = m+aµγµ+bµγ5γµ+ 1Hµνσµν, (2) Cj′ = −mj +(c0j +cj0)+ej, (6)
2
b
0
D = δ (d +d δ )
where the superscripts e and p are to be added to the j′k m jk− kj 00 jk
symbolsaµ,bµ,cµν,dµν,eµ,fµ,gλµν,andHµν thatrepre- 1 1
sent tensors encoding Lorentz violation for the fermions. εklm( gmlj +gm00δjl) εjklHl0, (7)
− 2 − m
3
1 +(D φp) Dµφp (mp)2(φp) φp (10)
Ej′k = −cjk− 2c00δjk, (8) 1 µ † 1 − †
F = (d +d ) 1 bj +d + 1ε g (9) −4FµνFµν − 4(kF)κλµνFκλFµν.
j′kl 0j j0 − 2 m j0 2 jmn mn0
(cid:20) (cid:18)
Ithasaconventionalprotonsectorandnon-conventional
1 1 b 1
l
+ ε H δ + + ε g δ electron and photon sectors. Lorentz violation for the
jmn mn kl lmn mn0 jk
2m 2 m 2
(cid:19)(cid:21) (cid:18) (cid:19) electronis givenby the coefficient kµν. As usual, the co-
−εjlm(gm0k+gmk0). variantderivativeisgivenbyDµφe,p =(∂µ+iqe,pAµ)φe,p,
where qe,p is the particle’s electric charge. If one iden-
tifies k = ce , this Lagrangian leads to the modified
µν µν
2. Interaction terms
Hamiltonian Eq. (3) if only the c are nonzero.
µν
For simplicity, consider the special case of only one
In addition to δh, the Hamiltonian arising from nonzerocomponentk =k2 1,wherekdeviatesslightly
00
the Lagrangian Eq. (1) also involves modified from unity. The lagrangiant−akes the form [18]
interaction terms proportional to combinations of
aµ,bµ,cµν,dµν,eµ,fµ,gλµν, and Hµν. For the non- Lφ = (Dµφe)†Dνφe+(k2−1)|D0φe|2−m2e(φe)†φe
relativistic electrons in solids, however, these are sup-
pressed by a factor given by α, the fine-structure con- +(Dµφp)†Dµφp−(mp)2(φp)†φp (11)
1 1
stant, relative to the modifications of the free-particle FµνF (k ) FκλFµν.
µν F κλµν
Hamiltoninian, Eq. (3) [23]. This is basically because −4 − 4
thetypicalenergyscaleforsuchelectronsistheRydberg
By coordinate transformations t tk, ~x ~x, the field
energy mα2/2. We can therefore neglect the modified → →
redefinition A A , A A k, and rescaling the elec-
interaction terms. 0 → 0 i → i
tric charge q q/k, one obtains the Lagrangian
→
φ = (D φe) D φe m2(φe) φe+(D φp) Dµφp
C. Coordinate and field redefinitions L µ † ν − † µ †
+(k 2 1)D φp 2 (mp)2(φp) φp (12)
− 0 †
− | | −
Someoftheparameterscontainedineitherthephoton, 1 1 1 k2
FµνF (k ) FκλFµν + − B2,
electron,orprotonsectorsofthe LagrangianEq. (1)can −4 µν − 4 F κλµν 2
beabsorbedintotheothersectorsbycoordinateandfield
where B is the magnetic field. Thus, the Lorentz viola-
redefinitions without loss of generality. Thus, not all of
tion in the electronsector has been movedto the proton
the coefficients in the Lagrangianhave separate physical
andphotonsectors. (If (k ) =0,the parameter1/k
meanings. Loosely speaking, in experiments where one F κλµν
can be interpreted as a modified velocity of light [18].)
compares the sectors against each other only differential
However,itisingeneralnotpossibletoeliminateLorentz
effects are meaningful.
violation in more than one sector at the same time.
For example, in a hypothetical world containing only
photons and electrons, the nine components of (k ) Cavity experiments compare the velocity of a light
F κλµν
not constrained by astrophysical experiments could be waveto a length defined by a crystal. In the light of Eq.
absorbed into the nine symmetric components of ce (11), the Lorentz violation for the electron acts via the
[6, 17, 18, 26]. By definition, either the photon µoνr term(k2 1)D0φe 2. Withatime-independentCoulomb
− | |
the electron sector could be taken as conventional with potential A0 = const., this contributes a term
all the Lorentz violation in the other sector. For ex-
2im(k2 1)[φ +q (A )φ] (13)
ample, for tests of Lorentz violation for the photon ,0 0
− − ℜ
[13,15,16,17,18],oneimplicitlyassumesaconventional
to the equation of motion for Φe = e imx0φe in the
electron sector. −
non-relativistic limit (obtained, in the usual way, by the
The presence of protons (and neutrons) in the solid
Euler-Lagrange equations and setting to zero terms of
changes this picture. We can still assume that one of
order A 2 and m0). denotes the real part. The sec-
the sectorsis conventional,but then in generalthe other | 0| ℜ
ond term modifies the coupling of the electron to the
sectors are Lorentz-violating. Choosing a conventional
Coulomb potential, causing a geometry change of the
protonsectorallowsus to disregardthe protonterms. It
crystal. Thus, a combination of k2 1 and the modi-
also fixes the definition of coordinates and fields so that −
fied velocity of light given by (k ) is measured in
the componentsofc cannotbeabsorbedinto(k ) F κλµν
µν F κλµν
the experiment.
in general, i.e. they acquire separate physical meanings.
In the alternative description by Eq. (12), the same
To illustrate this, it suffices to consider an extension
Lorentz violation acts via a term analogous to Eq. (13)
of the toy versionof the SME introduced in [18]. Its La-
intheequationforthe proton,i.e.,arescaledcouplingof
grangian describes electrons and protons as scalar fields
φe and φp, neglecting spin effects: the proton to the electric field, and a modified velocity
of light given by (k ) and k2 1. Physically, both
F κλµν
φ = (η +k )(D φe) D φe m2(φe) φe pictures are equivalent. −
µν µν µ † ν †
L −
4
Here,weconsideredonlyasingleparameteranalogous the superscripts p denotes parameters for the proton) to
to c , that causes a scaling of the solid that is rotation- about 10 27GeV [31]. Comparing the frequencies of hy-
00 −
invariant. Thus, it cannot be measured in usual cavity drogenmasers,[32]find B′pmp+B m <2 10 28GeV
experiments, that search for a modulation of the effect (m and mp are the elec|trJon and pJ′rot|o∼n m×ass,−respec-
connected to a rotation of the cavity in space. How- tively).
ever, the tensors cµν emerging from our special case via The potential for further tests of Lorentz invariance
the three Lorentz boosts can — i.e., at least three out in space is discussed in [33]; for the electron, limits on
of nine degrees of freedom contained in the symmetric severalparametercombinationsareexpectedusing133Cs
part of cµν. It is not impossible that, by coordinate and and 87Rb clocks. However,these tests allow no limits on
field redefinitions, some of the other parameters can be the components of E .
j′k
absorbed into quantities that have no measurable effect.
However, as we have shown, at least 12 out of 18 pa-
rameters from the photon and electron sector (restrict- III. INFLUENCE OF δh ON SOLIDS
ing the electron sector to those parameters that are not
constrained by cosmological experiments) would be sep-
To first order in the changes, the influence of Lorentz
arately measurable, that can e.g. be chosen as three c
µν violationintheelectron’sequationofmotionontheprop-
and nine (k ) .
F κλµν erties of a crystal is induced by the expectation values
Inwhatfollows,weadoptaconventionalprotonsector,
ofthe Lorentz-violatingcontributiontothe hamiltonian,
with allthe Lorentz violationinthe electronandphoton
thatiscalculatedinthissection. InSec. IIIA,wepresent
sectors. Onecouldpossiblyextractthemeasurablequan-
ouransatzfortheelectronwavefunction;theexpectation
tities from c and consider only those in what follows;
µν value δh is then calculated in Sec. IIIB.
however,thereby one would single out a preferred frame Wehdeniote by (x ) , (p ) , and (σ )i the spatial, 3-
a i a i a
inwhichthemeasurablequantitiesaredefined,andloose
momentum,andPaulimatricesfortheathparticle. The
the covariance under observer Lorentz transformations
non-relativistic single-particle hamiltonian for the ath
which otherwise holds in the SME. Therefore, we choose particle is denoted h = hˆ +δh . The hamiltonian of
a a a
nottodosoandtreatallelementsofc asindependent.
µν the solid
1
h = hˆ +δh + hˆ +δh (15)
D. Previous experimental limits on electron all a a 2 a,b a,b
parameters Xa h i Xa6=bh i
is the sum of hˆ +δh over all particles, plus the sum of
a a
It is convenient to express limits on the coefficients the interaction terms hˆ over all pairs, and over δh ,
a,b a,b
withinasun–centeredcelestialequatorialreferenceframe
a possible Lorentz-violatingcorrection to it. (The factor
asdefinedin[23]. The componentsofquantitiesgivenin 1 correctsforthedouble-countingofpairs.) TheLorentz
that frame are denoted by capital indices. Limits for 2
violating terms are contained in δh and δh . To first
a a,b
many particles, including muons, protons, and neutrons,
order in the changes, the resulting modifications of the
havebeenstudied,see[22,23,27]andreferencestherein.
properties of the solid are the sum of the modifications
Forthe electron,the limitsgivenbelowhavebeenfound.
arising form the individual terms.
However, to our knowledge there are no experimental
The interactions in a solid are electromagnetic. The
limits on E and on many components of F for the
j′k j′kl geometrychangeofcrystalsasaconsequenceofthemod-
electron.
ificationtotheinteractiontermformthe photonicsector
From clock comparison experiments [23], a limit on
ofthe SME [17, 18] has been treatedfor ionic crystalsin
BJ <10−24 (mBJ isdenoted˜bJ in[23])isobtained. Fur- [19]. We will not consider this term further here. In this
ther∼more, for the linear combinations work, we deal with the modifications due to the Lorentz
d˜ = m(d +d ) 1 md + 1ε H , violation in the electrons’ equation of motion, aδha.
J 0J J0 J0 JKL KL
− 2 2
(cid:18) (cid:19) P
1
g˜ = mε (g + g ) b , (14) A. Wave function ansatz for the solid
D,J JKL K0L KL0 J
2 −
d˜J/m<10−19 and g˜D,J/m<10−19. These are order-of- According to the Bloch theorem ([34], pp. 133-141),
magnit∼ude limits, since som∼e assumptions are needed to thesingle-electronwavefunctionψ forthea-thelectron
a
extract them from the measurements [23]. (a = 1,...,N) of a solid can be written as the prod-
An experiment using spin polarized solids yielded uct of a plane wave exp i~q ~x (where ~q is the quasi-
~ a a a
|BZ′ | ≃ (2.7 ± 1.6) × 10−25 [28, 29]; in a similar ex- momentumoftheathele{ctron)}andafunctionuq~(~r)with
periment [30], ((B )2 + (B )2)1/2 6.0 10 26 and the period of the lattice. u (~r) depends on~q , and thus
B 1.4 10 25X′have beenY′ found.≤ × − on the electron number a.q~Tao make a Fourieraexpansion
| Z′ |≤ × −
Hydrogenspectroscopycouldprospectivelylimitlinear ofu (~r),wenotethat,ifk isthe3 3matrixcontaining
combinationsofB ,B′p,d ,dp ,H ,andHp (where theq~parimitive reciprocallatjtiice vecto×rs~k , any reciprocal
J′ J J0 J0 JK JK i
5
lattice vector can be expressed as a linear combination vanishes. Thecaseofspinpolarizationwillbeconsidered
n k with some coefficients n Z ([34], pp. 86-87). in the appendix. Furthermore, we assume a vanishing
i ji i
∈
Therefore, sum of the helicities of the electrons,
1 i
ψa = √V exp ~~qa~xa (c~n)aexp{−injkij(xa)i}, (pa)j(σa)k =0 (22)
(cid:26) (cid:27)Xn~ and Xa (cid:10) (cid:11)
(16)
where V is the volume of the (macroscopic) solid con-
sidered. The (c~n)a are the Fourier coefficients of uq~a(~r); (pa)i(pa)j(σa)k =0. (23)
they depend only on n k (q ) , i.e., they can be ex- a
j ij − a i X(cid:10) (cid:11)
pressed as c . The ~n summation is carried out
over Z3. Thnej(kcij−)(qas)aitisfy Althoughsituations couldbe imaginedwhichviolate the
n a lasttworelationsinspiteofSl =0,thiscanbeconsidered
highly unrealistic. Therefore, the expectation values of
(c ) 2 =1 (17)
| ~n a| the terms proportional to Cj, Djk, and Fj′kl from the
~n
X hamiltonian, Eq. (3), vanish. Disregarding the constant
because of normalization (ψ) (ψ) = 1. If we assume term proportional to A,
a a
h | i
thatthecrystalhasinversionsymmetry,theoriginofthe
coordinate system may be chosen such that ([34], p.137) 1 N
(δh) = E (p ) (p ) .
h i m i′j h a i a ji
(c ~n)a =(c~n∗)a. (18) Xa=1
−
The star denotes complex conjugation.
The normalized antisymmetric N-electron state ΨN
2. Calculation of δh
(N is the total number of electrons) can be constructed
from the N N matrix
× We now calculate the matrix element (p ) (p ) for
1 i 1 j
h i
ψ (~x ) ... ψ (~x ) the first electron. Since it turns out to be independent
1 1 1 N
(ψ):= ... ... ...  (19) fsryommmtehtreyeolefcttrhoenNs n-eulmecbtreorn(astactoensΨeqNu)e,ntcheeosfutmheofanthtie-
ψ (~x ) ... ψ (~x )
 N 1 N N  matrixelementsforallelectronscanthenbeobtainedby
  multiplying (p ) (p ) with the total number of elec-
as the Slater determinant [35] ΨN = 1 det(ψ) [det de- h 1 i 1 ji
√N! trons N. We have
notes the determinant of a square matrix].
~2∂2
(p1)i(p1)j = ΨN∗ − ΨNd3x1...d3xN.
h i ∂(x ) (x )
B. Calculation of matrix elements ZV 1 i 1 j
(24)
The integrations are carried out over the volume V of
1. Specifications thesolid. ΨN andΨN aregivenbySlaterdeterminants.
∗
Evaluationof the matrix element starts by an expansion
For a bound system in its rest frame, the expectation of these determinants with respect to the first column,
value of the particle momenta vanishes
N
h(pa)ii=0. (20) det(ψ)= (−1)adet(|1ψ∗a), (25)
Xa Xa=1
We also assume no spin polarization,i.e., the sum of the a
where( ψ )denotesthe(N 1) (N 1)minormatrix
spin expectation values |1 ∗ − × −
obtainedfromtheN N matrix(ψ)bydeletingthefirst
×
column and the a-th row. The derivatives can then be
1
Sl (σ )l (21) carried out:
a
≡ N
a
X(cid:10) (cid:11)
N
1
(p ) (p ) = ( 1)a+b (c ) (c ) (q ) (q ) +~2n k n k ~n k (q ) ~n k (q ) (26)
h 1 i 1 ji N! − m~ ∗a ~n b b i b j l il k jk− k ik b j − l jl b i
a,b=1 ~n,m~
X X (cid:2) (cid:3)
6
1 exp i(x ) (m n )k + 1[(q ) (q ) ] d3x det( ψ a)det( ψb)d3x ...d3x .
×V 1 l i− i li ~ b l− a l 1 |1 ∗ |1 2 N
ZV (cid:26) (cid:20) (cid:21)(cid:27) ZV
Note that only the first integral in Eq. (26) contains~x1. 3. Estimating ξlk
Using the abbreviation
1 The properties of the wave function enter the momen-
κ =(m n )k + [(q ) (q ) ], (27)
l i− i li ~ b l− a l tum expectation value via ξlk and qiqj.
ξ is an ensemble average over all electrons:
the d3x -integration in Eq. (26) can be expressed as lk
1
ZV exp{i(x1)lκl}d3x1 =(cid:26)V0 ((κκll 6==00))., (28) ξlk = ~n N1 a |(c~n)a|2nlnk!=: ~n |c~n|2nlnk, (35)
X X X
Since the quasi-momenta (q ) are within the first
a i
Brillouin zone ([34], p. 89), (q ) (~/2)n k for obtained by substituting Eq. (30) into Eq. (32). A de-
a l j lj
any nj Z 0 . Thus, κl =| 0 on|l≤y if ni =| mi|and tailedevaluationofξlk wouldstartfrommaterialspecific
(qb)l = (∈qa{)l.\W}e may assume that this holds only for Fourier coefficients c~n obtained experimentally or the-
a=b. That allowsto carryoutthe m~ andthe b summa- oretically (see, e.g, [37]). Since detailed wave-function
tions. We now use calculations for realistic materials are notoriously diffi-
cult and would have to be performed for each individual
(c ) 2n =0, (29)
| ~n a| i material,itisinterestingtousearelativelysimplemodel
X~n for the Fourier coefficients. Such a model might already
which follows from Eq. (18) and eliminates the terms be quite accurate, since only the averageof the absolute
linear in n from the first line of Eq. (26) [36]. We define square c 2 is required, rather than the detailed (c )
~n ~n a
| |
for the individual electrons. A (possibly complicated)
(ξ ) := (c ) 2n n (30)
a lk ~n a l k dependency of the (c ) on ~n can be hoped to smooth
| | ~n a
X~n out in the averaging. The model must, however, be in
and accordance with the requirement that the wave function
1 has the rotational symmetry of the lattice, i.e, cρˆ~n c~n
qiqj := (qa)i(qa)j, (31) if ρˆ is any operator of the rotation group of the cry≡stal
N
a ([37], pp. 469).
X
theaverageofthequasi-momentumproductqiqj overall For such a simple model, we assume that c~n c~n ,
electrons. We also define the average i.e., the averageof the absolute squaresof th|e a|ve≡ra|g|e|o|f
1 the Fourier coefficients depends only on ~n. It follows
ξij := (ξa)ij. (32) that | |
N
a
X
Together with Eq. (17), this yields ξlk =γmatδlk (36)
(p ) (p ) = N q q +~2ξ k k with some material dependent constant γmat. For deter-
1 i 1 j i j lk il jk
h i N! mining γ , we note that the average kinetic energy of
mat
× d3x2...d3xN(cid:2)det(|1ψ∗a)det(|1ψb(cid:3)) (33) aanndel(e3c6t)r,on hTi = 21mhpipjiδij. From Eqs. (34), (35),
ZV
= q q +~2ξ k k ΨN 1 ΨN 1 .
i j lk il jk − | − ~2
T =γ δ k k δ . (37)
Toproveth(cid:2)at ΨN−1 ΨN−1 (cid:3)=(cid:10) 1 weexpand(cid:11)the remain- h i mat2m lk il jk ij
| a
ingdeterminantsof( ψ )intermsofthecolumnwhich
(cid:10) |1 ∗ (cid:11) If we neglect for the moment the energy of the chemical
is now the first one. The d3x integration can then be
2 bonding,this shouldcorrespondto the averageofthe ki-
carried out. The procedure is repeated, until the d3x
N netic energies of the atoms’ valence electrons, which can
integration is done. Each step reduces the dimension
be estimated using the Bohr model. The kinetic energy
of the Slater determinant by one and produces a factor
ofanelectronintheBohrmodelT =E Z/n2. Here,
Bohr R
equal to the number of electrons still involved. Taking
E 13.6eV is the Rydberg energy, Z the charge num-
R
allthesefactorstogethercancelsthenormalizationfactor ≃
ber ofthe atomcore,and n the principal quantumnum-
1/(N 1)!. Thus, we obtain the desired result
− ber. Thus,thekineticenergyfromtheBohrmodel,aver-
(p ) (p ) =q q +~2ξ k k . (34) agedovertheelectronswithintheatomsofthemolecule,
1 i 1 j i j lk il jk
h i
Since the right hand side of this result contains no refer- 1 Ne,m E Z
encetotheelectron’snumber,itholdsforalla=1,...,N T = v R k (38)
electrons. Bohr Ne,m k=1 k n2k
X
7
where N gives the number of valence electrons per Thequasi-momenta ~q arerestrictedtothefirstBril-
e,m a
molecule. The index k enumerates the atoms of the louin zone, ~q < ~ ~k|. |Most electrons, however, will
molecule and vk,nk, and Zk are the valence, principal haveaquasi-|m|omen2t|ujm| lowerthanthismaximumvalue,
quantum number, and charge number, respectively, of sothat the averageq q is a relativelysmallcontribution
i j
the atom k. Note that Zk = vk, since in an atom with to δh . If we neglect it, we obtain the final result
v valence electrons,the chargenumber ofthe atomcores h i
~2
and the inner shell electrons is Z = v. For example, in
δh =Nγ E δ k k . (41)
quartz,SiO , there is one Si atomwith v =4, n=3 and h i matm i′j lk il jk
2
two O atoms with v =2 and n=2. Thus, for example,
δ k k is symmetric in the indices i and j. This result
lk il jk
willbeusedinthe nextsectiontocomputethegeometry
E 42 22 17
(T ) = R 1 +2 = E . (39) change of the crystal caused by Lorentz violation in the
Bohr quartz 8 32 22 36 R electrons’ equation of motion.
(cid:18) (cid:19)
Comparing this to Eq. (37), we obtain the material spe-
cific values γmat given in Tab. I. IV. CHANGE OF CRYSTAL GEOMETRY
The model can be refined by taking into account the
energy of the chemical bonding, which leads to an in- The direct lattice vectors ~l contained in the matrix
a
crease of the actual kinetic energy of the electrons. The
l determinethestructureofthelatticewithoutLorentz
socalledenthalpy offormation∆fH0 givesthe enthalpy viiaolation. Lorentzviolationwillcauseachange˜l ofthe
ia
fortheformationofthecrystalfromtheelementsintheir
crystalgeometry,i.e.,thelatticevectorsarenowgivenby
usual state at standard conditions (room temperature l +˜l . Tocalculateit,weadjust˜l suchastominimize
ia ia ia
and pressure), e.g., solid or diatomic (O , for example).
2 the total energy of the lattice
The Bohrmodel,however,predictsthe energyoftheun-
boundatoms. Thatmeans,thechangeT−TBohrbetween U =U0(lia+˜lia)+ δh(lia+˜lia) . (42)
the sum of the kinetic energy of the valence electrons of
D E
the free atoms TBohr and the sum of their kinetic en- The first term is the conventional total energy of the
ergyinthe moleculeT is the difference of∆fH0 andthe lattice without Lorentz violation. It can be expressed as
applicable enthalpies of sublimation ∆ H0 or dissoci-
ation ∆dissH0 of the elements. For sapspubhlire, Al2O3, for U0(lia+˜lia)=U0(lia)+Uelast(˜lia)+Uc (43)
example,
where U is a constant and U is the elastic energy
c elast
connected to a distortion of the lattice. If ˜l = 0, the
∆ H0(Al O ) = 16.8eV ia
f 2 3 elastic energy U = 0. The total energy is thus given
∆ H0(2Al) = 2 3.0eV elast
− subl − × by
∆ H0(3O) = 3 2.5eV
diss
− − ×
T −TBohr = 3.3eV, U =U0(lia)+Uelast(˜lia)+ δh(lia+˜lia) . (44)
D E
or about 5% of TBohr = 68eV. This would lead to a 5% The correction˜lia is found by setting to zero the deriva-
increaseofthefactorγmat. Thisindicatesthattheenergy tive:
of the chemical bonding is, for our purposes, negligible.
The ultimate refinement of the model would be the ∂U ∂U ∂ δh(lia+˜lia)
elast
insertion of material specific Fourier coefficients c into = + =0. (45)
~n ∂˜l ∂˜l D ∂˜l E
Eq. (35). The precision of the model would then ap- jb jb jb
proach the limitations of the Bloch ansatz for the wave To explicitly calculate ˜l , we have to express the contri-
jb
function itself, which is based on a mean field model for butions to U in terms of ˜l . We will do so for δh in
the electron-electroninteractions. Such adetailedmodel jb h i
Sec. IVA and in Sec. IVB for U . In Sec. IVC, the
is, however,beyond the scope of the present work. elast
totalenergyperunitcellthusobtainedisminimizedand
the geometry change as expressed by a strain tensor e
ij
is calculated.
4. Result
The expectation value of the Lorentz-violatingcorrec- A. Dependence of δh on ˜lia
tion to the single particle hamiltonian δh can be
a a
obtained from Eq. (34) by multiplying with the number The change of the hamiltonian’s expectation value
P
of electrons. Inserting ξlk as obtained in the previous δh , Eq. (41) depends on the geometry via the recip-
h i
section into Eq. (34), we obtain rocal lattice vectors k , for which we have the relation
ij
([34], p. 87),
N
hδhi= mEi′j qiqj +γmat~2δlkkilkjk . (40) lijkik =2πδjk, (46)
(cid:0) (cid:1)
8
and therefore l k = 2πδ . If we substitute l +˜l with some coefficients η N. Multiplying this equation
ij nj in ij ij a
∈
and kjk +k˜jk, with ˜lij lij and k˜jk kjk, we obtain by kbc and using Eq. (46), we obtain
≪ ≪
(l +˜l )(k +k˜ )=2πδ (47) 1
ij ij ik ik jk η = k x . (56)
c bc b
2π
or
If l is shifted to l +˜l , the lattice point originally at
ba ba ba
lijkik+lijk˜ik+˜lijkik+˜lijk˜ik =2πδjk. (48) xb will be shifted to xb+ub, where
Thefirsttermonthel.h.s. cancelswithther.h.sdueEq. 1
u =η ˜l = k x ˜l . (57)
(46); we neglect the second order term on the l.h.s., and b a ba 2π da d ba
obtain
Therefore, we have
l k˜ +˜l k =0. (49)
ij ik ij ik
∂u 1
d = k ˜l (58)
Multiplying with knj and using Eq. (46) again, ∂xc 2π ca da
1 or
k˜ = k ˜l k . (50)
nk nj ij ik
−2π
1
e = k ˜l +k ˜l . (59)
We can now substitute k +k˜ into Eq. (41) to obtain dc 4π da ca ca da
ij ij
δh(˜l ): (cid:16) (cid:17)
ab This can now be used to express the elastic energy in
~2 1 terms of ˜lab:
δh(˜l ) = Nγ δ k k ˜l k
ab mat lk il ia ba bl
m − 2π 1 1 1
(cid:18) (cid:19) U = λ (k ˜l +k ˜l ) (k ˜l +k ˜l )V .
1 elast 2 ijkl4π ia ja ja ia 4π lb kb kb lb
× kjk − 2πkja˜lbakbk Ei′j (51) (60)
(cid:18) (cid:19) Somemanipulationofindicesusingλ =λ =λ
N~2 abcd bacd cdab
= const γ E (52) leads to the more simple form
− mat2πm i′j
×(kia˜lbakbkkjk+kja˜lbakbkkik). Uelast = 8Vπ2λijklkia˜ljaklb˜lkb. (61)
Theconst=Nγ ~2E δ k k doesnotdependon˜l .
matm i′j lk il jk ab
A term of order E (˜l )2 has been neglected.
i′j ab C. Minimizing the total energy per unit cell
B. Elastic energy Summing up the contributions, we find for the energy
change per unit cell (leaving out the constant terms)
The elastic energy is given as [38]
1 N~2
U = det(l ) λ k ˜l k ˜l γ (62)
1 | ij |8π2 ijkl ia ja lb kb− mat2πm
U = λ e e V , (53)
elast 2 ijkl ij kl E (k ˜l k k +k ˜l k k )
× i′j ia ba bk jk ja ba bk ik
where λ is the elastic modulus, V the volume consid-
ijkl were V = det(l ), the volume of a unit cell, and
ered, and | ij |
N = N the number of valence electrons per unit cell
e,u
1 ∂ui ∂uj havebeeninserted(nottobeconfusedwithNe,m thecor-
eij = + (54) responding number per molecule). The inner-shell elec-
2 ∂x ∂x
(cid:18) j i(cid:19) trons are assumed not to influence the crystal geometry.
is the strain tensor, where u is the displacement of a A minimum is found, when
i
volumeelementatsomelocationx . Fori=j,u repre-
i ij
∂U
sentstherelativechangeoflengthinxi-direction,andfor =0. (63)
i=j, itrepresentsthe changeofthe rightanglebetween ∂˜l
mn
6
lines originally pointing in x and x direction.
i j
To express the elastic energy in terms of ˜l , we note Aftersomemanipulation,thederivativecanbeexpressed
ij
as
thatthelocationx ofapointofthedirectlatticecanbe
b
expressedasa linearcombinationofthe primitivelattice ∂U det(l ) N ~2
vectors ∂˜l = | 4π2ij |λimklkinklb˜lkb−γmat 2eπ,um (64)
mn
x =η l , (55) E (k k k +k k k )=0.
b a ba × i′j in mk jk jn mk ik
9
We denote E = 1(E +E ) the symmetric part of is reduced for a symmetric crystal. For a compact pre-
(′ij) 2 i′j j′i
the tensor E . The last equation can be simplified a bit sentation of the material specific results in the following
i′j
by multiplying with l : sections, we will arrange these into a 6 6 matrix, that
pn ×
allows to express Eq. (69) as a 6 dimensional matrix
4π~2N equation
λ k ˜l γ e,u E k k =0. (65)
pmkl lb kb− mat det(l )m (′pj) mk jk
| ij | eΓ =BΓΞEΞ′ . (74)
For solving this for ˜l, we need the inverse µ (called
abkl Wethereforearrangethesixindependentelementsofe
dc
the compliance tensor) of λ , defined by
abcd and E(′bj) as the vectors
µ λ =δ δ . (66)
abkl abcd kc dl e = (e ,e ,e ,e ,e ,e ), (75)
Γ xx yy zz yz zx xy
Multiplying Eq. (65) with µdepm gives EΓ′ = (Ex′x,Ey′y,Ez′z,Ey′z,Ez′x,Ex′y) (76)
4πN ~2 [the capitalgreek indices run from 1...6] and define the
keb˜ldb =γmat det(le,u) mE(′pj)µdempkmkkjk (67) sensitivity matrix
ij
| |
2 2 2
and a further multiplication by les yields B1111 B1122 B1133 B1123 B1131 B1112
2 2 2
B2211 B2222 B2233 B2223 B2231 B2212 
2~2N 2 2 2
˜lds =γmatmdet(eli,uj) E(′pj)µdempleskmkkjk. (68) B =B23331111 B23332222 B23333333 2B23332233 2B23333311 2B23331122  .
| | B B B B B B 
 2 2 2 
Thestraintensorcanbecalculatedfromthisresultusing B3111 B3122 B3133 B3123 B3131 B3112 
 2 2 2 
Eq. (59) as B1211 B1222 B1233 B1223 B1231 B1212 
 (77)
e = ˜ E (69) The factors of 2 account for the double-counting of the
dc Bdcpj (′pj) non-diagonalelementsofE inthetensorequationEq.
(′pj)
with (69).
2N ~2
˜ =γ e,u µ k k . (70)
Bdcpj matmdet(l ) dcmp mk jk 2. Sensitivity matrix for isotropic materials
ij
| |
ThishasbeensimplifiedbyusingEq. (46). SinceforEq. Let us first consider isotropic materials that have no
(69), this is multiplied with the symmetric E(′jp), only preferred crystal orientation, i.e. crystals of cubic struc-
the part = 1(˜ + ˜ ) that is symmetric in j ture and non–crystalline (fused) materials which consist
Bdcpj 2 Bdcpj Bdcjp
and p of a large number of small crystals oriented statistically.
Cubic materials have one single lattice constant a; the
Ne,u~2 matrix of the primitive direct lattice vectors is given by
=γ (µ k k +µ k k )
Bdcjp matmdet(lij) dcmp mk jk dcmj mk pk lij = aδij. According to Eq. (46), the matrix of the re-
| | (71) ciprocal lattice vectors is given by kij = (2π/a)δij. In
is used. The resulting strain tensor the appendix, it is described how to obtain the compli-
ance constants µ from the elasticity constants that
abcd
e = E (72) are tabulated for various materials in the literature, e.g.
dc Bdcpj p′j
[39]. InsertingintoEqs. (71,77),weobtainthesensitivity
is given by the 3×3 Lorentz violation tensor Ep′j and a matrix B. It is of the structure
tensor , which gives the sensitivity of the material
dcpj
geometBrychangetoLorentzviolationintheelectronsec- 11 12 12 0 0 0
B B B
tor of the SME. This is the desired resultof this section. 0 0 0
B12 B11 B12 
0 0 0
=B12 B12 B12  . (78)
B  0 0 0 44 0 0 
1. General properties of the sensitivity tensor Bdcpj and  B 
 0 0 0 0 0 
conventions  B44 
 0 0 0 0 0 
 B44 
 
The sensitivity tensor has the symmetries
For cubic crystals, the non-zero values
Bdcjp =Bcdjp =Bdcpj. (73) 11 = ξ/(a2)C11,
B
In general, however, = . It has, therefore, 12 = ξ/(a2)C12,
at most 36 independeBntdcejlpem6 enBtjsp,dtche number of which B = ξ/(2a2)C ,
44 44
B
10
where
TABLEI:Elementsofthesensitivitymatrixforfusedand/or
8π2N ~2 cubic materials. fq denotes fused quartz, fs fused sapphire,
e,u
ξ =γmat (79) C denotes diamond. Materials for which three elements of B
mdet(l )
| ij | aregivenareisotropic;thecoefficientsshouldbeinsertedinto
Eq. (78). The other materials are trigonal; the coefficients
and C are the elements of the 6 6 compliance ma-
ΓΞ
× for these are to be inserted into Eq. (82).
trix, Eq. (B9). From symmetry arguments, this is also
the structure of the matrix for non crystalline materi- Mat. γmat B11 B12 B13 B14 B31 B33 B41 B44
B
als without a preferredorientation. The elements of this fq 0.38 0.77 -0.09 - - - - - 0.57
matrix for some cubic and/or fused materials are given fs 0.29 0.06 -0.01 - - - - - 0.05
intableI. Thevaluesforfusedquartzandsapphirehave
Si 0.50 2.51 -0.70 - - - - - 2.05
been calculated from the values of the crystalline ma-
C 1.16 5.77 -0.59 - - - - - 5.35
terials (to be calculated below) as averages over crystal
Al2O3 0.29 0.14 -0.01 -0.00 0.02 -0.02 0.01 0.01 0.04
orientations.
SiO2 0.38 1.41 -0.07 -0.06 0.35 -0.14 0.44 0.25 0.41
3. Sensitivity matrix for trigonal crystals
= ξ[1/(3a2)+1/(2c2)]C ,
14 14
B
Quartz and Sapphire are of trigonal structure and are 31 = 2ξ/(a2)C13,
B
frequentlyusedincavityexperiments. Therefore,wealso = ξ/(c2)C ,
33 33
B
consider the trigonal case here. The matrix of the prim- = ξ/(a2)C ,
41 14
itive direct lattice vectors can be chosen as B
= ξ[1/(6a2)+1/(4c2)]C .
44 44
B
a/2 a/2 0
C are the elements of the compliance matrix given in
lij = √3a/2 −√3a/2 0  (80) EqΓΞ. (B9). Numerical values of ΓΞ for quartz and sap-
0 0 c B
phire are given in Tab. I. The matrix is not sym-
 
  metrical; the elements of the first columnBare generally
whereaandcarethetwolatticeconstants. Wecalculate
the highest in this matrix, i.e., the geometry change of
the inverse k ; the product
ij trigonal materials is most sensitive to the xx element of
the Lorentz violation parameters E . This is because
2/a2 0 0 (′ij)
the direct lattice vector components in x-direction are a
kikkjk =4π2 0 2/(3a2) 0  (81) factor √3 smaller than the y components. Hence, the
0 0 1/c2 wave function of the electrons oscillate faster in x direc-
 
  tion, i.e. the px momentumcomponentis larger. Since
turns out to be a diagonal matrix. Trigonal crystals the influencehof iLorentz violation is given by the p p
i j
have six independent compliance constants and two lat- matrix element, this means a higher influence of thhe xi-
tice constants, which makes 8 independent components component of the Lorentz violation coefficients E .
(′ij)
for the -matrix. It has the structure
High elastic constants decrease the values of so that
B
B
crystals of high stiffness (such as sapphire) should show
0 0
B11 B12 B13 B14 lower values of . However, in some cases (particularly
3 1 0 0 B
 B12 3B11 B13 −B14  diamond), this is outweighed by small dimensions of the
B = BB3411 −1331BB3141 B033 B044 00 00  (82) udnuiettcoelslh(othrtatpiemripodlyohfigthhemeolemcetrnotnumwaevxepefuctnacttiioonnsv)alauneds
 0 0 0 0 2  a large number of electrons per unit cell.
 B55 3B41 
 0 0 0 0 
 B55 B66 
 
with V. EXPERIMENTS
1
55 = 44+ 41, Here, we discuss the application of our results to ex-
B B 3B
tract limits on Lorentz violation in the electrons’ equa-
1
= . tion of motion from experiments.
66 11 12
B 3B −B
The matrix elements are explicitly
A. Lorentz violation signal in cavity experiments
= 2ξ/(a2)C ,
11 11
B
= 2ξ/(3a2)C ,
12 12 Asdiscussedintheintroduction,Lorentzviolationmay
B
= ξ/(c2)C , affect the resonance frequency of a cavity ν through a
13 13
B