Download 2. Non-relativistic field theories

EMFT (ELECTROMAGNETIC FIELD THEORY)
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1. A classical field theory is a physical theory that describes the study of how one or more physical
fields interact with matter. The word 'classical' is used in contrast to those field theories that incorporate
quantum mechanics (quantum field theories).
A physical field can be thought of as the assignment of a physical quantity at each point
of space and time. For example, in a weather forecast, the wind velocity during a day over a country is
described by assigning a vector to each point in space. Each vector represents the direction of the
movement of air at that point. As the day progresses, the directions in which the vectors point change as
the directions of the wind change. From the mathematical viewpoint, classical fields are described by
sections of fiber bundles (covariant classical field theory). The term 'classical field theory' is commonly
reserved for describing those physical theories that describe electromagnetism and gravitation, two of
the fundamental forces of nature.
Descriptions of physical fields were given before the advent of relativity theory and then revised in light of
this theory. Consequently, classical field theories are usually categorised as nonrelativistic and relativistic.
2. Electromagnetism
Historically, the first (classical) field theories were those describing the electric and magnetic fields
(separately). After numerous experiments, it was found that these two fields were related, or, in fact, two
aspects of the same field: the electromagnetic field. Maxwell's theory of electromagnetism describes the
interaction of charged matter with the electromagnetic field. The first formulation of this field theory used
vector fields to describe the electric and magnetic fields. With the advent of special relativity, a better (and
more consistent with mechanics) formulation using tensor fields was found. Instead of using two vector
fields describing the electric and magnetic fields, a tensor field representing these two fields together is
used.
We have the electromagnetic potential,
, and the electromagnetic four-
current
. The electromagnetic field at any point in spacetime is described by the
antisymmetric (0,2)-rank electromagnetic field tensor
[edit]The Lagrangian
To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we
have
us
We can use gauge field theory to get the interaction term, and this gives
[edit]The Equations
This, coupled with the Euler-Lagrange equations, gives us the desired result, since the E-L
equations say that
It is easy to see that
. The left hand side is trickier. Being careful with
factors of
, however, the calculation gives
the equations of motion are:
. Together, then,
This gives us a vector equation, which are Maxwell's equations in vacuum. The other
two are obtained from the fact that F is the 4-curl of A:
where the comma indicates a partial derivative.
3. Relativistic field theory
Modern formulations of classical field theories generally require Lorentz covariance as this is now
recognised as a fundamental aspect of nature. A field theory tends to be expressed mathematically by
using Lagrangians. This is a function that, when subjected to an action principle, gives rise to the field
equations and a conservation law for the theory.
We use units where c=1 throughout.
Lagrangian dynamics
Given a field tensor
constructed from
, a scalar called the Lagrangian density
can be
and its derivatives.
From this density, the functional action can be constructed by integrating over spacetime
Therefore the Lagrangian itself is equal to the integral of the Lagrangian Density over all space.
Then by enforcing the action principle, the Euler-Lagrange equations are obtained
20 marks:
1.Relativistic fields
Two of the most well-known Lorentz covariant classical field theories are now described.
Electromagnetism
Historically, the first (classical) field theories were those describing the electric and magnetic fields
(separately). After numerous experiments, it was found that these two fields were related, or, in fact, two
aspects of the same field: the electromagnetic field. Maxwell's theory of electromagnetism describes the
interaction of charged matter with the electromagnetic field. The first formulation of this field theory used
vector fields to describe the electric and magnetic fields. With the advent of special relativity, a better (and
more consistent with mechanics) formulation using tensor fields was found. Instead of using two vector
fields describing the electric and magnetic fields, a tensor field representing these two fields together is
used.
We have the electromagnetic potential,
, and the electromagnetic four-
current
. The electromagnetic field at any point in spacetime is described by the
antisymmetric (0,2)-rank electromagnetic field tensor
[edit]The Lagrangian
To obtain the dynamics for this field, we try and construct a scalar from the field. In the vacuum, we
have
us
We can use gauge field theory to get the interaction term, and this gives
[edit]The Equations
This, coupled with the Euler-Lagrange equations, gives us the desired result, since the E-L
equations say that
It is easy to see that
. The left hand side is trickier. Being careful with
factors of
, however, the calculation gives
the equations of motion are:
. Together, then,
This gives us a vector equation, which are Maxwell's equations in vacuum. The other
two are obtained from the fact that F is the 4-curl of A:
where the comma indicates a partial derivative.
[edit]Gravitation
Main articles: Gravitation and General Relativity
After Newtonian gravitation was found to be inconsistent with special
relativity, Albert Einstein formulated a new theory of gravitation called general
relativity. This treats gravitation as a geometric phenomenon ('curved spacetime')
caused by masses and represents the gravitational field mathematically by a tensor
field called the metric tensor. The Einstein field equations describe how this
curvature is produced. The field equations may be derived by using the EinsteinHilbert action. Varying the Lagrangian
,
where
tensor
is the Ricci scalar written in terms of the Ricci
and the metric tensor
, will yield the vacuum field equations:
,
where
is the Einstein tensor.
2. Non-relativistic field theories
Some of the simplest physical fields are vector force fields. Historically, the first time fields were taken
seriously was with Faraday's lines of force when describing the electric field. Thegravitational field was
then similarly described.
Newtonian gravitation
A classical field theory describing gravity was Newtonian gravitation, which describes the gravitational
force as a mutual interaction between two masses.
In a gravitational field, if a test particle of gravitational mass m experiences a force F, then
the gravitational field strength 'g' is defined by "g = F/m", where it is required that the test mass, m, be so
small that its presence effectively does not disturb the gravitational field. Newton's law of gravitation says
that two masses separated by a distance, r, experience a force
where is a unit vector pointing away from the other object. Using Newton's 2nd law (for constant
inertial mass), F=ma leads to a definition of the gravitational field strength due to a mass m as
The experimental observation that inertial mass and gravitational mass are equal
to unprecedented levels of accuracy leads to the identification of the gravitational field strength
as identical to the acceleration experienced by a particle. This is the starting point of
the equivalence principle, which leads to general relativity.
Electrostatics
A charged test particle, charge q, experiences a force, F, based solely on its charge. We can
similarly describe the electric field, E, so that F=qE. Using this and Coulomb's law tells us that,
we define the electric field due to a single charged particle as
Modern formulations of classical field theories generally require Lorentz covariance as this is
now recognised as a fundamental aspect of nature. A field theory tends to be expressed
mathematically by using Lagrangians. This is a function that, when subjected to an action
principle, gives rise to the field equations and a conservation law for the theory.
We use units where c=1 throughout.
Lagrangian dynamics
Given a field tensor
density
, a scalar called the Lagrangian
can be constructed from
and its derivatives.
From this density, the functional action can be constructed by integrating over spacetime
Therefore the Lagrangian itself is equal to the integral of the Lagrangian Density over all
space.
Then by enforcing the action principle, the Euler-Lagrange equations are obtained
Relativistic fields
Two of the most well-known Lorentz covariant classical field theories are now
described.
Electromagnetism
Historically, the first (classical) field theories were those describing the electric and
magnetic fields (separately). After numerous experiments, it was found that these
two fields were related, or, in fact, two aspects of the same field:
the electromagnetic field. Maxwell's theory of electromagnetism describes the
interaction of charged matter with the electromagnetic field. The first formulation of
this field theory used vector fields to describe the electric and magnetic fields. With
the advent of special relativity, a better (and more consistent with mechanics)
formulation using tensor fields was found. Instead of using two vector fields
describing the electric and magnetic fields, a tensor field representing these two
fields together is used.
We have the electromagnetic potential,
, and
the electromagnetic four-current
. The electromagnetic field at any
point in spacetime is described by the antisymmetric (0,2)-rank electromagnetic field
tensor
The Lagrangian
To obtain the dynamics for this field, we try and construct a scalar from the
field. In the vacuum, we have
theory to get the interaction term, and this gives us
We can use gauge field
The Equations
This, coupled with the Euler-Lagrange equations, gives us the desired
result, since the E-L equations say that
It is easy to see that
. The left hand side is
trickier. Being careful with factors of
gives
motion are:
, however, the calculation
. Together, then, the equations of
This gives us a vector equation, which are Maxwell's equations in
vacuum. The other two are obtained from the fact that F is the 4curl of A:
where the comma indicates a partial derivative.
[edit]Gravitation
Main articles: Gravitation and General Relativity
After Newtonian gravitation was found to be inconsistent
with special relativity, Albert Einstein formulated a new theory
of gravitation called general relativity. This
treats gravitation as a geometric phenomenon
('curved spacetime') caused by masses and represents
the gravitational field mathematically by a tensor field called
the metric tensor. The Einstein field equations describe how
this curvature is produced. The field equations may be derived
by using the Einstein-Hilbert action. Varying the Lagrangian
,
where
is the Ricci scalar written in terms
of the Ricci tensor
and the metric tensor
, will
yield the vacuum field equations:
,
where
tensor.
is the Einstein
3. History
Quantum field theory originated in the 1920s from the problem of creating a quantum mechanical
theory of the electromagnetic field. In particular de Broglie in 1924 introduced the idea of a wave
description of elementary systems in the following way: "we proceed in this work from the assumption of
the existence of a certain periodic phenomenon of a yet to be determined character, which is to be
attributed to each and every isolated energy parcel".[2] In 1925, Werner Heisenberg, Max Born,
and Pascual Jordan constructed such a theory by expressing the field's internal degrees of freedom as an
infinite set of harmonic oscillators and by employing the canonical quantization procedure to those
oscillators. This theory assumed that no electric charges or currents were present and today would be
called a free field theory. The first reasonably complete theory of quantum electrodynamics, which
included both the electromagnetic field and electrically charged matter (specifically, electrons) as
quantum mechanical objects, was created by Paul Dirac in 1927.[3] This quantum field theory could be
used to model important processes such as the emission of a photon by an electron dropping into
a quantum state of lower energy, a process in which the number of particles changes—one atom in the
initial state becomes an atom plus a photon in the final state. It is now understood that the ability to
describe such processes is one of the most important features of quantum field theory.
It was evident from the beginning that a proper quantum treatment of the electromagnetic field had to
somehow incorporate Einstein's relativity theory, which had grown out of the study of classical
electromagnetism. This need to put together relativity and quantum mechanics was the second major
motivation in the development of quantum field theory. Pascual Jordan and Wolfgang Paulishowed in
1928 that quantum fields could be made to behave in the way predicted by special
relativity during coordinate transformations (specifically, they showed that the
field commutators wereLorentz invariant). A further boost for quantum field theory came with the
discovery of the Dirac equation, which was originally formulated and interpreted as a single-particle
equation analogous to the Schrödinger equation, but unlike the Schrödinger equation, the Dirac equation
satisfies both the Lorentz invariance, that is, the requirements of special relativity, and the rules of
quantum mechanics. The Dirac equation accommodated the spin-1/2 value of the electron and accounted
for its magnetic moment as well as giving accurate predictions for the spectra of hydrogen. The attempted
interpretation of the Dirac equation as a single-particle equation could not be maintained long, however,
and finally it was shown that several of its undesirable properties (such as negative-energy states) could
be made sense of by reformulating and reinterpreting the Dirac equation as a true field equation, in this
case for the quantized "Dirac field" or the "electron field", with the "negative-energy solutions" pointing to
the existence of anti-particles. This work was performed first by Dirac himself with the invention of hole
theory in 1930 and by Wendell Furry, Robert Oppenheimer, Vladimir Fock, and others. Schrödinger,
during the same period that he discovered his famous equation in 1926, also independently found the
relativistic generalization of it known as the Klein-Gordon equation but dismissed it since, without spin, it
predicted impossible properties for the hydrogen spectrum. (See Oskar Klein and Walter Gordon.) All
relativistic wave equations that describe spin-zero particles are said to be of the Klein-Gordon type.
Of great importance are the studies of Soviet physicists, Viktor Ambartsumian and Dmitri Ivanenko, in
particular the Ambarzumian-Ivanenko hypothesis of creation of massive particles (published in 1930)
which is the cornerstone of the contemporary quantum field theory.[4] The idea is that not only the quanta
of the electromagnetic field, photons, but also other particles (including particles having nonzero rest
mass) may be born and disappear as a result of their interaction with other particles. This idea of
Ambartsumian and Ivanenko formed the basis of modern quantum field theory and theory of elementary
particles.[5][6]
A subtle and careful analysis in 1933 and later in 1950 by Niels Bohr and Leon Rosenfeld showed that
there is a fundamental limitation on the ability to simultaneously measure the electric and magnetic field
strengths that enter into the description of charges in interaction with radiation, imposed by
the uncertainty principle, which must apply to all canonically conjugate quantities. This limitation is crucial
for the successful formulation and interpretation of a quantum field theory of photons and electrons
(quantum electrodynamics), and indeed, any perturbative quantum field theory. The analysis of Bohr and
Rosenfeld explains fluctuations in the values of the electromagnetic field that differ from the classically
"allowed" values distant from the sources of the field. Their analysis was crucial to showing that the
limitations and physical implications of the uncertainty principle apply to all dynamical systems, whether
fields or material particles. Their analysis also convinced most people that any notion of returning to a
fundamental description of nature based on classical field theory, such as what Einstein aimed at with his
numerous and failed attempts at a classical unified field theory, was simply out of the question.
The third thread in the development of quantum field theory was the need to handle the statistics of
many-particle systems consistently and with ease. In 1927, Jordan tried to extend the canonical
quantization of fields to the many-body wave functions of identical particles, a procedure that is
sometimes called second quantization. In 1928, Jordan and Eugene Wigner found that the quantum field
describing electrons, or other fermions, had to be expanded using anti-commuting creation and
annihilation operators due to the Pauli exclusion principle. This thread of development was incorporated
into many-body theory and strongly influenced condensed matter physics and nuclear physics.
Despite its early successes quantum field theory was plagued by several serious theoretical difficulties.
Basic physical quantities, such as the self-energy of the electron, the energy shift of electron states due to
the presence of the electromagnetic field, gave infinite, divergent contributions—a nonsensical result—
when computed using the perturbative techniques available in the 1930s and most of the 1940s. The
electron self-energy problem was already a serious issue in the classical electromagnetic field theory,
where the attempt to attribute to the electron a finite size or extent (the classical electron-radius) led
immediately to the question of what non-electromagnetic stresses would need to be invoked, which would
presumably hold the electron together against the Coulomb repulsion of its finite-sized "parts". The
situation was dire, and had certain features that reminded many of the "Rayleigh-Jeans difficulty". What
made the situation in the 1940s so desperate and gloomy, however, was the fact that the correct
ingredients (the second-quantized Maxwell-Dirac field equations) for the theoretical description of
interacting photons and electrons were well in place, and no major conceptual change was needed
analogous to that which was necessitated by a finite and physically sensible account of the radiative
behavior of hot objects, as provided by the Planck radiation law.
This "divergence problem" was solved in the case of quantum electrodynamics during the late 1940s and
early 1950s by Hans Bethe, Tomonaga, Schwinger, Feynman, and Dyson, through the procedure known
as renormalization. Great progress was made after realizing that all infinities in quantum electrodynamics
are related to two effects: the self-energy of the electron/positron and vacuum polarization.
Renormalization concerns the business of paying very careful attention to just what is meant by, for
example, the very concepts "charge" and "mass" as they occur in the pure, non-interacting fieldequations. The "vacuum" is itself polarizable and, hence, populated by virtual particle (on shell and off
shell) pairs, and, hence, is a seething and busy dynamical system in its own right. This was a critical step
in identifying the source of "infinities" and "divergences". The "bare mass" and the "bare charge" of a
particle, the values that appear in the free-field equations (non-interacting case), are abstractions that are
simply not realized in experiment (in interaction). What we measure, and hence, what we must take
account of with our equations, and what the solutions must account for, are the "renormalized mass" and
the "renormalized charge" of a particle. That is to say, the "shifted" or "dressed" values these quantities
must have when due care is taken to include all deviations from their "bare values" is dictated by the very
nature of quantum fields themselves.
The first approach that bore fruit is known as the "interaction representation", (see the article Interaction
picture) a Lorentz covariant and gauge-invariant generalization of time-dependent perturbation theory
used in ordinary quantum mechanics, and developed by Tomonaga and Schwinger, generalizing earlier
efforts of Dirac, Fock and Podolsky. Tomonaga and Schwinger invented a relativistically covariant
scheme for representing field commutators and field operators intermediate between the two main
representations of a quantum system, the Schrödinger and the Heisenberg representations (see the
article on quantum mechanics). Within this scheme, field commutators at separated points can be
evaluated in terms of "bare" field creation and annihilation operators. This allows for keeping track of the
time-evolution of both the "bare" and "renormalized", or perturbed, values of the Hamiltonian[disambiguation
needed] and expresses everything in terms of the coupled, gauge invariant "bare" field-equations.
Schwinger gave the most elegant formulation of this approach. The next and most famous development
is due to Feynman, who, with his brilliant rules for assigning a "graph"/"diagram" to the terms in the
scattering matrix (See S-Matrix Feynman diagrams). These directly corresponded (through
the Schwinger-Dyson equation) to the measurable physical processes (cross sections, probability
amplitudes, decay widths and lifetimes of excited states) one needs to be able to calculate. This
revolutionized how quantum field theory calculations are carried-out in practice.
Two classic text-books from the 1960s, J.D. Bjorken and S.D. Drell, Relativistic Quantum
Mechanics (1964) and J.J. Sakurai, Advanced Quantum Mechanics (1967), thoroughly developed the
Feynman graph expansion techniques using physically intuitive and practical methods following from
the correspondence principle, without worrying about the technicalities involved in deriving the Feynman
rules from the superstructure of quantum field theory itself. Although both Feynman's heuristic and
pictorial style of dealing with the infinities, as well as the formal methods of Tomonaga and Schwinger,
worked extremely well, and gave spectacularly accurate answers, the true analytical nature of the
question of "renormalizability", that is, whether ANY theory formulated as a "quantum field theory" would
give finite answers, was not worked-out till much later, when the urgency of trying to formulate finite
theories for the strong and electro-weak (and gravitational interactions) demanded its solution.
Renormalization in the case of QED was largely fortuitous due to the smallness of the coupling constant,
the fact that the coupling has no dimensions involving mass, the so-called fine structure constant, and
also the zero-mass of the gauge boson involved, the photon, rendered the small-distance/high-energy
behavior of QED manageable. Also, electromagnetic processes are very "clean" in the sense that they
are not badly suppressed/damped and/or hidden by the other gauge interactions. By 1958 Sidney
Drell observed: "Quantum electrodynamics (QED) has achieved a status of peaceful coexistence with its
divergences...".
The unification of the electromagnetic force with the weak force encountered with initial difficulties due to
the lack of accelerator energies high enough to reveal processes beyond the Fermi interaction range.
Additionally, a satisfactory theoretical understanding of hadron substructure had to be developed,
culminating in the quark model.
In the case of the strong interactions, progress concerning their short-distance/high-energy behavior was
much slower and more frustrating. For strong interactions with the electro-weak fields, there were difficult
issues regarding the strength of coupling, the mass generation of the force carriers as well as their nonlinear, self interactions. Although there has been theoretical progress toward a grand unified quantum
field theory incorporating the electro-magnetic force, the weak force and the strong force, empirical
verification is still pending. Superunification, incorporating the gravitational force, is still very speculative,
and is under intensive investigation by many of the best minds in contemporary theoretical physics.
Gravitation is a tensor field description of a spin-2 gauge-boson, the "graviton", and is further discussed in
the articles on general relativity and quantum gravity.
From the point of view of the techniques of (four-dimensional) quantum field theory, and as the numerous
and heroic efforts to formulate a consistent quantum gravity theory by some very able minds attests,
gravitational quantization was, and is still, the reigning champion for bad behavior. There are problems
and frustrations stemming from the fact that the gravitational coupling constant has dimensions involving
inverse powers of mass, and as a simple consequence, it is plagued by badly behaved (in the sense of
perturbation theory) non-linear and violent self-interactions. Gravity, basically, gravitates, which in
turn...gravitates...and so on, (i.e., gravity is itself a source of gravity,...,) thus creating a nightmare at all
orders of perturbation theory. Also, gravity couples to all energy equally strongly, as per the equivalence
principle, so this makes the notion of ever really "switching-off", "cutting-off" or separating, the
gravitational interaction from other interactions ambiguous and impossible since, with gravitation, we are
dealing with the very structure of space-time itself. (See general covariance and, for a modest, yet highly
non-trivial and significant interplay between (QFT) and gravitation (spacetime), see the article Hawking
radiation and references cited therein. Also quantum field theory in curved spacetime).
Thanks to the somewhat brute-force, clanky and heuristic methods of Feynman, and the elegant and
abstract methods of Tomonaga/Schwinger, from the period of early renormalization, we do have the
modern theory of quantum electrodynamics (QED). It is still the most accurate physical theory known, the
prototype of a successful quantum field theory. Beginning in the 1950s with the work of Yang and Mills,
as well as Ryoyu Utiyama, following the previous lead of Weyl and Pauli, deep explorations illuminated
the types of symmetries and invariances any field theory must satisfy. QED, and indeed, all field theories,
were generalized to a class of quantum field theories known as gauge theories. Quantum
electrodynamics is the most famous example of what is known as an Abelian[disambiguation needed] gauge
theory. It relies on the symmetry group U(1) and has one massless gauge field, the U(1) gauge symmetry,
dictating the form of the interactions involving the electromagnetic field, with the photon being the gauge
boson. That symmetries dictate, limit and necessitate the form of interaction between particles is the
essence of the "gauge theory revolution". Yang and Mills formulated the first explicit example of a nonAbelian gauge theory, Yang-Mills theory, with an attempted explanation of the strong interactions in mind.
The strong interactions were then (incorrectly) understood in the mid-1950s, to be mediated by the pimesons, the particles predicted by Hideki Yukawa in 1935, based on his profound reflections concerning
the reciprocal connection between the mass of any force-mediating particle and the range of the force it
mediates. This was allowed by the uncertainty principle. The 1960s and 1970s saw the formulation of a
gauge theory now known as the Standard Model of particle physics, which systematically describes the
elementary particles and the interactions between them.
The electroweak interaction part of the standard model was formulated by Sheldon Glashow in the years
1958-60 with his discovery of the SU(2)xU(1) group structure of the theory. Steven Weinberg and Abdus
Salam brilliantly invoked the Anderson-Higgs mechanism for the generation of the W's and Z masses (the
intermediate vector boson(s) responsible for the weak interactions and neutral-currents) and keeping the
mass of the photon zero. The Goldstone/Higgs idea for generating mass in gauge theories was sparked
in the late 1950s and early 1960s when a number of theoreticians (including Yoichiro Nambu, Steven
Weinberg, Jeffrey Goldstone, François Englert, Robert Brout, G. S. Guralnik, C. R. Hagen, Tom
Kibble and Philip Warren Anderson) noticed a possibly useful analogy to the (spontaneous) breaking of
the U(1) symmetry of electromagnetism in the formation of the BCS ground-state of a superconductor.
The gauge boson involved in this situation, the photon, behaves as though it has acquired a finite mass.
There is a further possibility that the physical vacuum (ground-state) does not respect the symmetries
implied by the "unbroken" electroweak Lagrangian (see the article Electroweak interaction for more
details) from which one arrives at the field equations. The electroweak theory of Weinberg and Salam
was shown to be renormalizable (finite) and hence consistent by Gerardus 't Hooft and Martinus Veltman.
The Glashow-Weinberg-Salam theory (GWS-Theory) is a triumph and, in certain applications, gives an
accuracy on a par with quantum electrodynamics.
Also during the 1970s, parallel developments in the study of phase transitions in condensed matter
physics led Leo Kadanoff, Michael Fisher and Kenneth Wilson (extending work of Ernst
Stueckelberg, Andre Peterman, Murray Gell-Mann, and Francis Low) to a set of ideas and methods
known as the renormalization group. By providing a better physical understanding of the renormalization
procedure invented in the 1940s, the renormalization group sparked what has been called the "grand
synthesis" of theoretical physics, uniting the quantum field theoretical techniques used in particle physics
and condensed matter physics into a single theoretical framework.
4. Principles of quantum field theory
Classical fields and quantum fields
Quantum mechanics, in its most general formulation, is a theory of abstract operators (observables)
acting on an abstract state space (Hilbert space), where the observables represent physically observable
quantities and the state space represents the possible states of the system under study. Furthermore,
each observable corresponds, in a technical sense, to the classical idea of adegree of freedom. For
instance, the fundamental observables associated with the motion of a single quantum mechanical
particle are the position and momentum operators and . Ordinary quantum mechanics deals with
systems such as this, which possess a small set of degrees of freedom.
(It is important to note, at this point, that this article does not use the word "particle" in the context
of wave–particle duality. In quantum field theory, "particle" is a generic term for any discrete quantum
mechanical entity, such as an electron or photon, which can behave like classical particles or classical
waves under different experimental conditions, such that one could say 'this "particle" can behave like a
wave or a particle'.)
A quantum field is a quantum mechanical system containing a large, and possibly infinite, number of
degrees of freedom. A classical field contains a set of degrees of freedom at each point of space; for
instance, the classical electromagnetic field defines two vectors — the electric field and the magnetic
field — that can in principle take on distinct values for each position r. When the field as a whole is
considered as a quantum mechanical system, its observables form an infinite (in fact uncountable) set,
because r is continuous.
Furthermore, the degrees of freedom in a quantum field are arranged in "repeated" sets. For example, the
degrees of freedom in an electromagnetic field can be grouped according to the position r, with exactly
two vectors for each r. Note that r is an ordinary number that "indexes" the observables; it is not to be
confused with the position operator encountered in ordinary quantum mechanics, which is an
observable. (Thus, ordinary quantum mechanics is sometimes referred to as "zero-dimensional quantum
field theory", because it contains only a single set of observables.)
It is also important to note that there is nothing special about r because, as it turns out, there is generally
more than one way of indexing the degrees of freedom in the field.
In the following sections, we will show how these ideas can be used to construct a quantum mechanical
theory with the desired properties. We will begin by discussing single-particle quantum mechanics and
the associated theory of many-particle quantum mechanics. Then, by finding a way to index the degrees
of freedom in the many-particle problem, we will construct a quantum field and study its implications.
[edit]Single-particle
and many-particle quantum mechanics
In quantum mechanics, the time-dependent Schrödinger equation for a single particle in one dimension is
where m is the particle's mass, V is the applied potential, and
denotes the wavefunction.
We wish to consider how this problem generalizes to N particles. There are two motivations for
studying the many-particle problem. The first is a straightforward need in condensed matter physics,
where typically the number of particles is on the order of Avogadro's number (6.0221415 x 1023). The
second motivation for the many-particle problem arises from particle physics and the desire to
incorporate the effects of special relativity. If one attempts to include the relativistic rest energy into
the above equation (in quantum mechanics where position is an observable), the result is either
the Klein-Gordon equation or the Dirac equation. However, these equations have many
unsatisfactory qualities; for instance, they possess energy eigenvalues that extend to –∞, so that
there seems to be no easy definition of a ground state. It turns out that such inconsistencies arise
from relativistic wavefunctions having a probabilistic interpretation in position space, as probability
conservation is not a relativistically covariant concept. In quantum field theory, unlike in quantum
mechanics, position is not an observable, and thus, one does not need the concept of a positionspace probability density. For quantum fields whose interaction can be treated perturbatively, this is
equivalent to neglecting the possibility of dynamically creating or destroying particles, which is a
crucial aspect of relativistic quantum theory. Einstein's famous mass-energy relation allows for the
possibility that sufficiently massive particles can decay into several lighter particles, and sufficiently
energetic particles can combine to form massive particles. For example, an electron and
a positron can annihilate each other to create photons. This suggests that a consistent relativistic
quantum theory should be able to describe many-particle dynamics.
Furthermore, we will assume that the N particles are indistinguishable. As described in the article
on identical particles, this implies that the state of the entire system must be either symmetric
(bosons) or antisymmetric (fermions) when the coordinates of its constituent particles are exchanged.
These multi-particle states are rather complicated to write. For example, the general quantum state of
a system of N bosons is written as
where
are the single-particle states, Nj is the number of particles occupying state j, and the
sum is taken over all possible permutations p acting on N elements. In general, this is a sum
ofN! (N factorial) distinct terms, which quickly becomes unmanageable as N increases. The way
to simplify this problem is to turn it into a quantum field theory.
Second quantization
In this section, we will describe a method for constructing a quantum field theory called second
quantization. This basically involves choosing a way to index the quantum mechanical degrees
of freedom in the space of multiple identical-particle states. It is based on
the Hamiltonian formulation of quantum mechanics; several other approaches exist, such as
the Feynman path integral,[7]which uses a Lagrangian formulation. For an overview, see the
article on quantization.
Second quantization of bosons
For simplicity, we will first discuss second quantization for bosons, which form perfectly
symmetric quantum states. Let us denote the mutually orthogonal single-particle states
by
and so on. For example, the 3-particle state with one particle in
state
and two in state
is
The first step in second quantization is to express such quantum states in terms
of occupation numbers, by listing the number of particles occupying each of the singleparticle states
etc. This is simply another way of labelling the states. For
instance, the above 3-particle state is denoted as
The next step is to expand the N-particle state space to include the state spaces for all
possible values of N. This extended state space, known as a Fock space, is composed
of the state space of a system with no particles (the so-called vacuum state), plus the
state space of a 1-particle system, plus the state space of a 2-particle system, and so
forth. It is easy to see that there is a one-to-one correspondence between the
occupation number representation and valid boson states in the Fock space.
At this point, the quantum mechanical system has become a quantum field in the sense
we described above. The field's elementary degrees of freedom are the occupation
numbers, and each occupation number is indexed by a number
of the single-particle states
, indicating which
it refers to.
The properties of this quantum field can be explored by defining creation and
annihilation operators, which add and subtract particles. They are analogous to "ladder
operators" in the quantum harmonic oscillator problem, which added and subtracted
energy quanta. However, these operators literally create and annihilate particles of a
given quantum state. The bosonic annihilation operator
and creation
operator
have the following effects:
It can be shown that these are operators in the usual quantum mechanical
sense, i.e. linear operators acting on the Fock space. Furthermore, they are
indeed Hermitian conjugates, which justifies the way we have written them.
They can be shown to obey the commutation relation
where stands for the Kronecker delta. These are precisely the relations
obeyed by the ladder operators for an infinite set of independent quantum
harmonic oscillators, one for each single-particle state. Adding or removing
bosons from each state is therefore analogous to exciting or de-exciting a
quantum of energy in a harmonic oscillator.
The Hamiltonian of the quantum field (which, through the Schrödinger
equation, determines its dynamics) can be written in terms of creation and
annihilation operators. For instance, the Hamiltonian of a field of free (noninteracting) bosons is
where
is the energy of the k-th single-particle energy eigenstate.
Note that
Hence,
eigenstate.
is known as the number operator for the k-th
[edit]Second quantization of fermions
It turns out that a different definition of creation and annihilation
must be used for describing fermions. According to the Pauli
exclusion principle, fermions cannot share quantum states, so
their occupation numbers Ni can only take on the value 0 or 1. The
fermionic annihilation operators c and creation operators
defined by their actions on a Fock state thus
are
These obey an anticommutation relation:
One may notice from this that applying a fermionic creation operator twice gives zero, so it is impossible
for the particles to share single-particle states, in accordance with the exclusion principle.
Field operators
We have previously mentioned that there can be more than one way of indexing the degrees of freedom
in a quantum field. Second quantization indexes the field by enumerating the single-particle quantum
states. However, as we have discussed, it is more natural to think about a "field", such as the
electromagnetic field, as a set of degrees of freedom indexed by position.
To this end, we can define field operators that create or destroy a particle at a particular point in space. In
particle physics, these operators turn out to be more convenient to work with, because they make it easier
to formulate theories that satisfy the demands of relativity.
Single-particle states are usually enumerated in terms of their momenta (as in the particle in a
box problem.) We can construct field operators by applying the Fourier transform to the creation and
annihilation operators for these states. For example, the bosonic field annihilation operator
.
is