Table Of ContentLecture Notes in Physics
Gaetano Lambiase
Giorgio Papini
The Interaction
of Spin
with Gravity in
Particle Physics
Low Energy Quantum Gravity
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Gaetano Lambiase (cid:129) Giorgio Papini
The Interaction of Spin
with Gravity in Particle
Physics
Low Energy Quantum Gravity
123
Gaetano Lambiase GiorgioPapini
Dipartimento di Fisica E.R.Caianiello Department ofPhysics
Universita’di Salerno University of Regina
Fisciano, SA,Italy Saskatchewan, SK,Canada
ISSN 0075-8450 ISSN 1616-6361 (electronic)
Lecture Notesin Physics
ISBN978-3-030-84770-8 ISBN978-3-030-84771-5 (eBook)
https://doi.org/10.1007/978-3-030-84771-5
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To Laura, Walter, my parents, my sister
and all friends for the constant support
and encouragement
Gaetano Lambiase
To Lucia, my sister Mariella, my sons
and everyone in my family for their patience,
encouragement and affection
Giorgio Papini
Preface
The behaviour of quantum systems in inertial and gravitational fields is of interest
ininvestigationsregardingthestructureofspace–timeatthequantumlevel.Though
a definitive answer to questions regarding the fundamental structure of space–time
may only come from a successful quantum theory of gravity, the extrapolation of
generalrelativityfromplanetarylengths,overwhichitiswellestablished,toPlanck
length requires a leap offaith in its validity of over forty orders of magnitude and
the resolution of difficult quantization problems. The alternative, performing
experiments at Planck length, appears, at present, remote. A more realistic
approach, limited to verifying Einstein’s theory at intermediate lengths, leads
necessarilytothestudyoftheinteractionofclassicalinertialandgravitationalfields
with quantum systems. These include, but are not limited to elementary particles.
AnumberofeffectscanbepredictedinthiswaywiththeadvantagethatEinstein’s
theoryaffordsaunifiedtreatmentofinertiaandgravity.Aprogrammeofthisnature
fulfils the double purpose of learning about the reaction of quantum systems to
gravity and also of confronting gravitational theory with the precision of quantum
measurements. Because of its weakness, gravitation is the fundamental interaction
least investigated at atomic and sub-atomic levels.
General relativity incorporates the equivalence principle from the outset and
observations, where feasible, do confirm that inertia and gravity interact with
quantum systems in ways that are compatible with Einstein’s views. This is borne
out of measurements on superconducting electrons and on neutrons which offer
tangibleevidencethattheeffectofinertiaandNewtoniangravityonwavefunctions
is that predicted by wave equations compliant with general relativity down to
lengths of 10(cid:1)5 cm and 10(cid:1)8 cm, respectively. General relativity is therefore
adhered to in the following, which would hardly require continued reference to
inertialfieldsasdistinctentities.Nonetheless,inertialeffectsmustbeidentifiedwith
great accuracy as dictated by their unavoidable presence in Earth-bound and
near-space experiments of ever increasing accuracy aimed at testing fundamental
theories.Inaddition,inertiaprovidesaguideinthestudyofrelativitybecause,inall
instanceswherenon-localityisnotatstake,theequivalenceprinciple,insomeofits
forms, ensures the existence of a gravitational effect for each inertial effect.
Quantum particles have mass, charge, and spin. Quantum spin arises from the
combination of special relativity with quantum mechanics. Every particle is
vii
viii Preface
associated with a particular field and every field corresponds to a class of indis-
tinguishable particles. These belong to two groups, fermions that have half integer
spin,andbosonswithintegerspin.Whilegravityinteractswithmassandchargein
auniversalway,itinteractswithspininwaysthatdependessentiallyonwhethera
particleisabosonorafermion.Spinplaysanimportantroleinparticlephysicsand
isinvolvedinsomeofthemostprecisefundamentalexperimentssofarperformed.
Suffice just to recall here the role of spin in the measurement of the anomalous
magnetic moment of the muon. Quantum spins are very versatile tools that can be
used in a variety of experimental situations and energy ranges while retaining an
essentially non-classical behaviour.
The issue of the interaction of quantum spin with gravity has been vigorously
pursued by Mashhoon who has argued that there should be a coupling of rotation,
hence gravity, to spin as an extension of the Sagnac effect that describes the
coupling of rotation to the angular momentum of a particle. It is with spin that the
difference between classical and quantum behaviour of a particle becomes partic-
ularly evident.
Fully covariant wave equations incorporate the spin characteristics of an object
fromtheoutset.Theypredicttheexistenceofaclassofinertial-gravitationaleffects
that may be tested experimentally in the not too distant future. In these equations
inertiaandgravityappearasexternalclassicalfields,but,byconformingtogeneral
relativity, provide valuable information on how Einstein’s views carry through in
the world of the quantum. The solutions of wave equations to first order in the
metricdeviationareexactandarecharacterizedbythefactthatgravityonlyappears
asaphaseinthewavefunctionandcanbecomputedbymeansofpathintegrals.It
is a Berry phase that can be understood as an effect that gravitation has on Hilbert
space. These gravitational Berry phases are derived in Chap. 1 for the most com-
monwaveequationsandexplicitlycalculated,inthefollowingchapters, inseveral
instances.
Phase differences are amenable to observation. Interferometers of large dimen-
sionsholdgreatpromiseinmanyoftheseinvestigations.Theycanprovideaccurate
measurementsofquantumphases,whoseroleisimportantingyroscopy,andinthe
testing of general relativity. It is anticipated that similar studies will be performed
with particle accelerators. The forerunner of this second group of investigations is
the work on the measurement of the muon’s magnetic moment in which evidence
can be found for the coupling of spin to rotation.
The spin-rotation coupling predicted by Mashhoon has been observed directly
for photons and neutrons, while applications to neutron interferometry are being
developed. This is a most significant development not only because the measure-
mentsconfirmtheexistenceofthephenomenon,butalsobecausetheyofferhintsof
deviation from that universality that people have come to expect of gravity.
Stillwithintheconfinesofspinphysics,theframedependentspinprecessionof
compound spin systems has become relevant in studies of heavy ions in storage
rings. Spintronics has also become a field of practical importance in which inertia,
rotational inertia in particular, has a role to play.
Preface ix
For most studies involving quantum particles the gravitational phase is of
paramountimportance.Forotherproblems,suchastheinteractionofgravitationat
the atomic level, the knowledge of the Hamiltonian is of greater importance. The
derivation of the Hamiltonian is usually accomplished by following a sequence of
Foldy–Wouthuysen transformations. It has been presented in comprehensive form
by Hehl and Ni purely within the framework of special relativity and in the local
frame of a fermion. In the present work, the non-relativistic case is tackled by
means of a procedure that renders the quantum phase manifest and can, therefore,
be applied to a larger class of problems. A high-energy approximation for Dirac
particleshasbeengivenbyCiniandTouschekin1958.Theirworkcanbeextended
to include external electromagnetic and gravitational fields and quantum phases to
any order. The derivation of the Hamiltonian can then be accomplished in a stan-
dard way. Use of some of these results is also made in the External Field
Approximation(EFA).ThegravitationalEFAhasdevelopedlessrapidlythaninthe
electromagnetic case because of the mathematically involved way in which grav-
itation is described in Einstein’s theory. It is, however, a useful tool in practical
applications. It is developed and applied in Chaps. 1 and 2.
Gravitational-inertial fields in the laboratory are weak and range from weak to
intermediatestrengthformostastrophysicalsources.Thesearethefieldsconsidered
here.TheyareadequatelydescribedbytheWeakFieldApproximation(WFA)and
can be applied, with caution, even in the neighbourhood of a strong source if
GM=c2r(cid:3)10(cid:1)1.
Of note among the applications are those to non-relativistic superconductors by
DeWitt, whose results were obtained when it was unknown whether general rela-
tivity could be applied in a quantum mechanical context. Results that applied to
non-inertial fields could find experimental confirmation, in an appropriate limit, in
the London moment of rotating superconductors.
Applicationstointerferometryasawaytotesttheactionofgravityonthewave
function of quantum systems are also given in Chap. 3.
Chapter4isdevotedtothebehaviourofneutrinosandtheirhelicityandflavour
oscillations in gravitational fields and to some consequences like spin coupling to
primordial gravitational waves and a pulsar’s increase in its rotation rate (pulsar
kick). Chapter 5 also deals with neutrinos, in particular with the effect of general
relativityonneutrinospinflavourandspin-fliposcillationsabovethecoreoftypeII
supernovae and off rotating black holes.
Some particular radiative processes are considered in Chap. 6 together with the
spin currents generated by non-inertial fields in which, unlike non-relativistic
particles, spin and orbital angular momenta are not separately conserved. Still in
Chap. 6, relativistic vortices are shown to arise as solutions of the covariant Dirac
equation.
Chapter7isconcernedwithaxionelectrodynamicsandthesearchforaxionsas
leading candidates for dark matter, with their influence on angular momentum and
spin of particles and with the formation of a chirally active space–time medium.
Current ideas on geometrical developments like torsion are also introduced.
x Preface
Chapter8dealswithperspectivesonseveralargumentstreatedinthebook.The
focus is mainly on the Gravitational Memory Effect and on the Quasi Normal
Modes, which could be detected with a future generation of gravitational wave
detectors. In addition, some important developments in holography, double-copy,
and Kawai–Lewellen–Tye relations, together with a recent proposal concerning
spin entanglement as a witness for quantum gravity are briefly discussed.
Spin-inertiaandspin-gravityinteractionsarethesubjectofnumeroustheoretical
and experimental efforts. An extensive, but hardly complete bibliography, is
attached.
Salerno, Italy Gaetano Lambiase
Toronto, Canada Giorgio Papini
June 2021
