Table Of ContentLarge Deviations Applied to Classical
and
Quantum Field Theory
Large Deviations Applied to Classical
and
Quantum Field Theory
Harish Parthasarathy
Professor
Electronics & Communication Engineering
Netaji Subhas Institute of Technology (NSIT)
New Delhi, Delhi-110078
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Preface
Thisbookdealswithavarietyofproblemsinphysicsandengineeringwhere
the Large deviation principle of probability finds application. Large deviations
is a branch of probability theory dealing with approximate computation of the
probabilities of rare events. Specifically we have a sequence of process valued
randomvariableswhichconvergestozeroortoaconstantastheparameterthat
indexestheprocessconvergestozeroortosomeotherconstant. Physicallythis
means that the parameter defines the degree of noise in the system that gener-
ates the process valued r.v. and in the limit when this parameter converges to
zero, thenoiseamplitudealsoconvergestozero. Insuchasituation, theproba-
bilitydistributionoftheprocessbecomesdegenerateinthelimitasforexample
happens in the weak law of large numbers. We are then interested in the rate
at which this probability distribution converges to the degenerate distribution.
Such a calculation then gives us the asymptotic probability of the rate event
that the process valued r.v. will assume values in a set that does not contain
the degenerate limit. The expression for this asymptotic probability of devia-
tion isdetermined by its ratefunction and isusually a much simpler expression
than that for the exact probability of deviation. Therefore we can use this rate
function to control the system parameters so that the deviation probability is
minimized. This is one application of the LDP to control theory. Other ap-
plications deal with electromagnetism, quantum mechanics, general relativity,
mechanics, cosmology, quantum field theory and quantum stochastic processes
and string theory which is the modern theory of quantum gravity. This book
analyzes a variety of such problems where the LDP can be applied. For ex-
ample in quantum mechanics and quantum field theory, we have an electron
boundtoitsnucleusonwhicharandomincidentelectromagneticradiationfield
is incident. When the random noise component in the field is small then we
can compute the effect of this component on the transition probability of the
electron between two of its stationary states and control the parameters in the
non-random field component so that this change in the transition probability is
a minimum. Such a computation would enable us to design a monochromatic
laser that is nearly insensitive to noisy perturbations. In general relativity,
we have for example the problem of controlling the background metric so that
the influence of random electromagnetic sources on the nature of gravitational
waves propagating in the background is minimized. This would enable us to
design more and more accurate tests for Einstein’s general theory of relativity.
In quantum field theory, we have for example the problem of controlling the
electron and photon propagators by applying external classical fields and cur-
rents to a system so that the effect of random noise in the environment on the
propagator deviations is minimized. This effect is measured by the probability
of deviation of the propagator function from the desired one in the absence of
the environmental noise. This would enable us to design more robust accelera-
tors for studying the S-matrix involving scattering, absorption and emission of
elementary particles. This book talks about several such examples. It contains
application s of the LDP to pattern recognition problems like for example anal-
ysis of the performance of the EM algorithm for optimal parameter estimation
v
inthepresenceofweaknoise,analysisandcontrolofnon-Abeliangaugefieldsin
thepresenceofnoise,quantumgravitywhereinweareconcernedwithperturba-
tiontothequadraticcomponentoftheEinstein-HilbertHamiltoniancausedby
higher order nonlinear terms in the position fields and their effect on the Gibbs
statisticsand consequently quantumprobabilitiesof eventscomputed usingthe
quantumGibbsstate. ThereaderwillalsofindinthisbookapplicationsofLDP
to quantum filtering theory as developed by Belavkin based on the celebrated
Hudson-Parthasarathy quantum stochastic calculus. The idea here is that the
estimateofthesystemdensityoperatorbasedonnon-demolitionmeasurements
follows an Abelian and hence classical stochastic dynamics and if the Lindblad
noise parameters in the HP equation are small, then we can in principle com-
pute the probability distribution of the filtered/estimated state process using
itsratefunctioninthelimitofzeroLindbladnoiseparameters. Applicationsto
string theory involve computing the change in the action functional of a point
field caused by its string theoretic evaluation followed by quantum averaging
with respect to the quantum fluctuating part of the string in a coherent state.
This idea also gives us a method to obtain string theoretic corrections to point
field theories. The book will be of use to graduate students in engineering and
mathematical physics as well as researchers in these fields.
Author
vi
Brief Contents
1. LDP Problems in Quantum Field Theory 1–26
2. LDP in Biology, Neural Networks, Electromagnetic
Measurements, Cosmic Expansion 27–34
3. LDP in Signal Processing, Communication and Antenna Design 35–42
4. LDP Applied to Quantum Measurement, Classical Markov Chains,
Quantum Stochastics and Quantum Transition Probabilities 43–68
5. LDP in Classical Stochastic Process Theory and
Quantum Mechanical Transitions 69–80
6. LDP in Pattern Recognition and Fermionic Quantum Filtering 81–106
7. LDP in Spin Field Theory, Anharmonic Perturbations of Quantum
Oscillators, Small Perturbations of Quantum Gibbs States 107–112
8. LDP for Electromagnetic Control of Gravitational Waves,
Randomly Perturbed Quantum Fields, Hartree-Fock
Approximation, Renewal Processes in Quantum Mechanics 113–124
9. LDP in Electromagnetic Scattering and String Theory,
Control of Dynamical Systems Using LDP 125–142
10. LDP in Markov Chain and Queueing Theory with Quantum
Mechanical Applications 143–147
11. LDP in Device Physics, Quantum Scattering Amplitudes,
Quantum Filtering and Quantum Antennas 149–154
12. How the Electron Acquires Its Mass, Estimating the Electron
Spin and the Quantum Electromagnetic Field Within a Cavity
in the Presence of Quantum Noise 155–162
13. Mathematical Tools for Large Deviations, Neural Networks,
LDP in Physical Theories, EM and LDP Algorithms
in Quantum Parameter Estimation and Filtering 163–170
14. Quantum Transmission Lines, Engineering Applications of
Stochastic Processes 171–186
15. More Tools in Probability, Electron Mass in the Presence of Gravity
and Electromagnetic Radiation, More on LDP in Quantum Field
Theory, Non-Abelian Gauge Field Theory and Gravitation 187–210
16. Weak Convergence, Sanov’s Theorem, LDP in
Binary Signal Detection 211–218
vii
17. LDP for Classical and Quantum Transmission Lines, String Theoretic
Corrections to Classical Field Lagrangians, Non-Abelian Gauge
Theory in the Language of D(cid:76)(cid:3445)(cid:72)(cid:85)(cid:72)(cid:81)(cid:87)(cid:76)(cid:68)(cid:79)(cid:3)Forms 219–228
18. LDP and EM Algorithm, LDP for Parameter Estimates in
Linear Dynamical Systems, Philosophical Questions in
Quantum General Relativity 229–237
19. Chapters Index 239–254
Detailed Contents
1. LDP Problems in Quantum Field Theory 1–26
1.1 Large Deviations for Supergravity Fields 1
1.2 Rate Function of String Propagator 2
1.3 Large Deviations for p-form Fields 5
1.4 The Dynamics of the Electro-weak Theory 7
1.5 Filtering in Fermionic Noise 7
1.6 Quantum Field Theory is a Low Energy Limit of String Field Theory 18
1.7 The Atiyah-Singer Index Theorem and LDP Problems Associated with It 21
1.8 LDP Problems in General Relativity 23
2. LDP in Biology, Neural Networks, Electromagnetic
Measurements, Cosmic Expansion 27–34
2.1 The Importance of Mathematical Models in Medicine 27
2.2 LDP Related Problems in Neural Networks and A(cid:85)(cid:87)(cid:76)(cid:191)(cid:70)(cid:76)(cid:68)(cid:79)(cid:3)Intelligence 30
2.3 LDP Related Problems to Cosmic Expansion in General Relativity 31
2.4 LDP Problems in Biology 31
2.5 A Sensitive Quantum Mechanical Method for Measuring
the Scattered Electromagnetic Fields 34
3. LDP in Signal Processing, Communication and Antenna Design 35–42
3.1 Review 35
3.2 Large Deviation Problems in SSB Modulation 36
3.3 What is Meant by Estimating a Quantum Field in Space-time 37
3.4 Write Down the Noisy Schrodinger Equation for an N Particle
System in the Formalism of Hudson and Parthasarathy and Derive
by Partial Tracing, the Approximate Nonlinear Stochastic Boltzmann
Equation for the State Evolution of a Single Particle (Evolution of
the Marginal Density with Noise) 38
3.5 The pde’s S(cid:68)(cid:87)(cid:76)(cid:86)(cid:191)(cid:72)(cid:71)(cid:3)(cid:69)(cid:92)(cid:3)(cid:87)(cid:75)(cid:72)(cid:3)Quantum Electromagnetic Field
Observables in a Cavity Resonator in the Presence of Bath Noise 40
3.6 Estimating the Quantum State of a Single Particle in a System of
N Indistinguishable Particles Interacting with Each Other and with
an External Bath Field 41
4. LDP Applied to Quantum Measurement, Classical Markov Chains,
Quantum Stochastics and Quantum Transition Probabilities 43–68
4.1 Some Other Aspects of Measurement of a Quantum Field 43
4.2 Some Parts of the Solution to the Question Paper on Antenna Theory 45
4.3 Some Additional LDP Related Problems in Antenna Theory 50
ix
