Quantum Theory and Relativity

Quantum Theory and Relativity
Arthur Jaffe∗
Harvard University
Cambridge, MA 02138, USA
Arthur Jaffe@harvard.edu
August 1, 2007
Dedicated to George Mackey
Abstract
Elsewhere I describe some personal encounters with George Mackey [31]. Those discussions
often returned to a common set of questions:
• Does physics suggest a way to combine quantum theory, relativity, and interaction?
• To what extent can one formulate such quantum physics as mathematics?
• What is the present status of quantum field theory?
• What new mathematics emerges?
• What new insight does one apparently require to make further progress?
We touch on these topics here, not only outlining some of the technical issues, but also attempt-
ing to address why one believes in their importance.
I
Introduction
Two major themes dominated twentieth century physics: quantum theory and relativity. These two
fundamental principles provide the cornerstones upon which one might build the understanding of
modern physics. And today after one century of elaboration of the original discoveries by Poincar´
e,
Einstein, Bohr, Schr¨
odinger, Heisenberg, Dirac—and many others—one still dreams of describing
the forces of nature within such an arena. Yet we do not know the answer to the basic question:
• Are quantum theory, relativity, and interaction mathematically compatible?
Even if one restricts relativity to special relativity, we do not know the answer to this question about
our four-dimensional world—much less about other higher-dimensional worlds considered by string
theorists!
Should quantum theory with relativity not qualify as logic? Physics suggests that a natural way
to combine quantum theory, special relativity, and interaction is through a non-linear quantum field.
Enormous progress on this problem has been made over the past forty years. This includes showing
that theories exist in space-times of dimension two and three. Building this new mathematical
framework and finding these examples has become known as the subject of constructive quantum
field theory.
We review some of these developments here. But before addressing this question further, we set
down a couple of ground rules. Although one cannot derive these starting points from basic first
principles, we attempt to explain the road we take.
∗I am grateful to the I.T.P. of the E.T.H. Z¨
urich for hospitality while I wrote this paper, and to Barbara Drauschke
for proof-reading the manuscript.
1

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2
Arthur Jaffe
Scope.
First let us ask about the domain of validity for the “laws of physics.” In searching for
these laws, can one be content in finding laws that are effective within a limited domain of nature—
such as laws that describe atoms and molecules, or laws that pertain to strong interactions and
electrodynamics, but that touch on neither gravity nor weak interactions? Or can one only be
satisfied when one can formulate a theory that encompasses all of physics?
For example, special relativity and the notion of Minkowski space-time revolutionized the notion
of classical physics. But one might argue that space-time itself cannot remain strictly Minkowski-
like at extremely high scales of energy or in the neighborhood of a black hole; so should one only
study theories where space-time can have a more general character—possibly curved, or possibly
with a quantum nature, or both? Can one justify, from the point of view of physics, the study of
special relativity in great detail? How much human thought and effort should one invest in giving
a mathematical foundation to an incomplete picture of nature?
Our position on this question is that past success in understanding physics has always been
partial and within limited domains. But within these domains one has pursued theories to their
logical conclusion. Insight into broad arenas of physics followed studies that ranged from Newton’s
laws of planetary motion, to extremal principles in the laws of mechanics and optics, Maxwell’s
laws of classical electromagnetism, probabilistic laws of Brownian motion, statistical foundations of
thermodynamics and phase transitions, and non-relativistic quantum theory—to mention just a few.
Yet each of these examples is specific to a certain domain of relevance. A big part of the success
and the appeal of these “theories” rests with their elegance and simplicity, and another appeal is the
depth of understanding they provide within each realm of physics. These theories not only model
parts of nature; but in addition they yield lasting beacons on the frontiers of knowledge—markers
to which one returns time and time again.
Of course one constantly strives to widen the scope of our understanding. As we acquire more
knowledge, we expand our horizons, always requiring that new points of view incorporate previous
successes and insights. Some physicists have proposed very ambitious plans; at one limit, string
theorists try to find a “theory of everything.” Laudable though this goal may be, the formulation
of all-encompassing principles has remained elusive.
With this as background, we state our first ground rule: it is valid to address the partial goal,
are quantum theory, special relativity and interaction compatible?
Logic.
If one accepts an intermediate and partial goal in the description of nature, then one
confronts a related question: why require that a partial theory have a firm foundation in logic?
In other words, why provide a valid mathematical basis for a partial set of physical laws? Is it
not better to focus effort on discovering new principles that describe a wider spectrum of natural
phenomena? If one accepts that view, then one might only attempt to clarify the logical foundations
of physics after achieving a much better, if still putative understanding of nature.
Our response to that proposal is another appeal to history.
Each of the classical areas of
physics mentioned under ‘scope’ qualifies as a subfield of mathematics. For centuries, the tradition
in physics has been to describe natural phenomena by mathematics. Eugene Wigner marveled
on the relevance of mathematics in his famous essay [60], “On the Unreasonable Effectiveness of
Mathematics in the Natural Sciences.” Intuition can go a long way. But by endowing physics with
a mathematical foundation, one also bestows physical laws with longevity. For mathematical ideas
can be understood and conveyed more easily than conjectures, both from person to person, and also
from generation to generation.
In recent years we have witnessed enormous progress in another direction—of transferring ideas
from physics to mathematics: to play on Wigner’s title, concepts from physics have had an un-
reasonable effectiveness in providing insight to formulate mathematical conjectures! The resulting
infusion of new perspectives has truly blossomed into a mathematical revolution, which has been
sufficiently robust to touch almost every mathematical frontier. I have written and talked on this

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Quantum Theory and Relativity
3
latter topic extensively elsewhere, but this is not the central theme that we investigate here.
Rather, we simply formulate the second ground rule: physics should be a subfield of mathematics!
In other words, one should add the adjective mathematically to the word compatible in ground
rule one. George Mackey often advocated this point of view in our conversations, and most every
mathematician would find this an appealing thought. Being mathematics does not limit the use of
insight or stop one making unjustifiable approximations to gain insight into a particular problem—
the only requirement is to be clear about what parts of physics are mathematics and which parts
are not.
The real question is whether this goal is realistic. No matter how laudable it may be to strive
for a mathematical theory of physics, can one expect to achieve it? This brings us inevitably to the
fundamental question posed at the outset. Understanding some self-contained, but complete nugget
in this sea of knowledge can produce work of lasting historical value—a mathematical work of art.
Framework.
We approach the question from the point of view of quantum field theory. In spite of
tremendous qualitative progress mentioned above in understanding space-time dimensions less than
four, the space-time in which we live still remains a mystery. So we first point out some facts that
motivate hope for the success of this quest, namely the success of quantum electrodynamics. Then
we explain some approaches to giving a mathematical foundation to quantum field theory. But we
then return to explain the currently-widespread belief that a mathematical treatment of quantum
electrodynamics may well be impossible!
If this were true, it would certainly represent a departure from traditional thinking—that one
can study the electromagnetic forces by themselves. But today physicists believe that one should
incorporate the equations of four-dimensional electrodynamics into the four-dimensional quantum
Yang-Mills equations, or elaborate them into the “standard model.”
In any case, the question of finding an interacting relativistic field in four dimensions appears
even more subtle than one might at first believe. For this and other reasons, we believe that the
question, “Does there exist a mathematically-complete, non-linear relativistic quantum field theory
in Minkowski four-space?” remains one of the most important unresolved questions in all of science.
II
Quantum Theory, Relativity, and Interaction
Here is the picture: States in quantum theory lie in a Hilbert space H. One assumes that there
is a unitary representation U of the Poincar´
e group G on H. Single particle states are defined as
subspaces K ⊂ H that carry an irreducible representation of this group. There is a distinguished
Poincar´
e-invariant state Ω ∈ H which represents the physical “vacuum.” A real quantum field
ϕ(x), when paired with a real, Schwartz-space function f (x), yields a symmetric linear operator
ϕ(f ) =
ϕ(x)f (x)dx acting on H. One says that ϕ is an operator-valued distribution.
Single particle states arise from the states ϕ(f )Ω. Multi-particle states occur in the application
of several fields to the vacuum, and the forces between particles are a consequence of the non-
linearity of the field equation. In this way, one can derive the force law between particles directly
from something fundamental—the form of the field equation. This non-linearity specifies the “laws
of physics.”
What other general features would one hope to incorporate? It is natural to expect that freely-
moving particles emerge asymptotically at large times from states of the field describing particles
moving away from one another. The fundamental properties of such states should arise from the
behavior of ϕ as a solution to a non-linear field equation. The particles would have fundamen-
tal properties as a consequence of the field equation, and this equation would also determine the
interaction and scattering of an arbitrary number of particles.
For example, a cubic non-linear interaction g ϕ3 for a scalar field with mass m, leads in lowest
3

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4
Arthur Jaffe
order perturbation theory to an attractive Yukawa-like force V (r) = g2 e−mr between two particles.
4πr
Basically the form of the force law in perturbation theory arises from the form of the Green’s
function for the time-independent linear equation. So one can ask whether this perturbation theory
consequence, as well as many others, can be translated into properties of an actual non-linear theory?
II.1
Compelling Evidence: Calculational Rules and Experiment
One can reflect on the great initial triumphs of quantum field theory, the prediction and measurement
of the Lamb shift in the spectrum of hydrogen and the value of the magnetic moment of the electron.
These events which began in 1947 gave credence to the picture described above and heralded the era
of the quantum field. They have been refined over sixty years to the point of unbelievable precision
between experiment and computation.
The greatest quantitative triumph of this perturbation picture comes from the rules developed
in quantum electrodynamics to calculate the magnetic moment of the electron. We focus on this,
because it has evolved into one of the most accurate agreements between theoretical calculation and
laboratory observation.
II.2
Quantum Electrodynamics
This year we celebrate the 60th-anniversary of the original calculation and observation of the de-
viation from its value that can be ascribed to the interaction between the electron (Dirac) field
ψ(x) and the electromagnetic (Maxwell) field F (x) = (E(x), B(x)). The heroic theoretical work of
Bethe, Weisskopf, Feynman, Schwinger, Tomonaga, Karplus, Kroll, and others led to the original
observation by Kusch in 1947.
Prior to 1947, there had been an evolution of understanding of this moment. A simple model
of the electron as a current loop with angular momentum J led to the picture that it would have a
magnetic moment µ = eJ/2mc. Taking the quantum value J = 1
for the angular momentum, this
2
meant the magnetic moment predicted by Bohr would be
e
µ Bohr =
.
(II.1)
4mc
In a magnetic field of magnitude B, the electron would have a potential energy µB, and this was
the original prediction of non-relativistic quantum theory.
The Dirac equation for a single relativistic electron naturally incorporates both special relativity
and the interaction with the magnetic field. It also predicts twice the interaction energy of the non-
relativistic theory that one could attribute to a magnetic moment of the electron. In other words,
the Dirac value of the magnetic moment would be
e
µ Dirac =
.
(II.2)
2mc
This was the value seen in observations.
However the proposal for a field theory explanation of the interaction between light and matter
led to a further refinement. The natural assumption about the form of the interaction introduced
the simplest non-linearity into the equations of quantum mechanics. The electron described by the
Dirac equation and the classical theory of light developed in the previous century by Maxwell could
be coupled similarly to the theory of classical electron motion in a magnetic field. One assumes that
the non-linear Lagrangian is
J · A dx, where the A(x) denotes the Maxwell potential and where
J (x) = e ¯
ψ(x)γψ(x) is the electric current. Here one takes J (x) as the same quadratic function of
the Dirac field ψ(x) that appeared in the original Dirac theory. The matrices γ are the famous

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Quantum Theory and Relativity
5
4 × 4 Dirac matrices; they satisfy the relation γµγν + γν γµ = 2gµν , similar to a Clifford algebra, but
incorporating the Minkowski metric gµν .
The problem with this procedure is that energy shifts due to the perturbation of the Lagrangian
for a linear wave equation turned out to be infinite. Eventually physicists devised a set of rules
to perform the calculations of observable effect. These rules involved taking infinite values of the
mass and charge in the original equations, while requiring that the resulting perturbed masses and
charges agree with observed values.
This non-linear field theory gave rise to rules of calculation in perturbation theory. One com-
puted that the electromagnetic field interacted with and modified the magnetic moment of the
electron. In 1947 Kusch, working at Columbia University, measured the change in the magnetic
moment. It agreed precisely with the calculated effect that is first-order in the square of the electric
charge, namely
µ Kusch = κµ Dirac ,
where κ = 1.001 .
(II.3)
This tiny increase of only 0.1 percent compared with the Dirac value could be measured, and ever
since one puts great credence in quantum field theory.
Over the intervening years, the measurements of µ and other related quantities has been refined
by many persons, especially Dehmelt and his student Gabrielse. Likewise the calculations have
undergone enormous progress. In order to take the accuracy to terms proportional to e4, one required
enormous computer power just to simplify the algebra required to multiply 4 × 4 matrices and
calculate the relevant traces, much less compute the many integrals of rational functions (Feynman
diagrams). A great expert who carried out much of the theoretical work is T. Kinoshita.
On this 60th-anniversary of the 1947 measurement, one can test the value of µ to unbeliev-
able accuracy. The latest result, see [43, 15], cited by the American Institute of Physics as the
“outstanding physics achievement of 2006,” gives
µ = κ60 µ Dirac ,
where κ60 = 1.001 159 652 180 85(±76) .
(II.4)
The calculations and theory agree completely to this extent. And the accuracy of this test astounds
the human mind. Thus while other physical theories (such as string theory or its ramifications) may
ultimately predict new observed phenomena, any such theory must also reproduce the predictions of
quantum electrodynamics including (II.4). Miraculously, these rules work in a precise quantitative
fashion, indicating that they might reflect a well-behaved underlying theory that we do not yet
understand.
III
Sense or Nonsense?
Faced in the early 1950’s with the astounding successes of the field concept in physics, as well as
the apparent robustness of the calculational rules, one could rejoice. However from a fundamental
point of view, quantum fields baffled all insight; what could be the meaning of equations of physics
that involve infinite constants and undefined operations? So we come back to the second question,
“To what extent can one formulate relativistic quantum physics with interaction as mathematics?”
Mathematicians and physicists achieved a love-hate relationship with quantum field theory.
Richard Feynman described renormalization, a subject that he and others pioneered, in terms of
a magician’s “hocus pocus,” see page 128 of [9]. Yet he also expressed a keen interest in knowing
whether relativistic quantum physics, and quantum electrodynamics in particular, has a mathemat-
ical foundation; see for example [8]. Many mathematicians held a distant view of quantum physics,
and not only because its foundations were imprecisely laid out. Many mathematicians, especially
those from the Bourbaki school, felt that mathematics itself needed to be put in order. They also
regarded modern physics as beyond the scope of mathematical understanding.

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Arthur Jaffe
Faced with this gulf in knowledge, a number of mathematicians and physicists attempted in
the 1950’s to formulate a mathematical framework in which quantum field theory might fit. These
efforts became known as Axiomatic QFT.
III.1
Axiomatic Quantum Field Theory
The early work, pioneered by Wightman, Jost, and Haag focused on formulating reasonable principles
one could require for any quantum field. The outline of this framework can be found in [56] with
elaboration in [57]. Full presentations and consequences of the axioms can be read in [53, 36].
One begins with the requirement that quantum theory satisfies some very basic properties which
have become known as the Wightman axioms. Basically they embody the following requirements:
• States lie in a quantum-mechanical Hilbert space H.
• There is a unitary, positive-energy representation U of the Poincar´
e group on H.
• There exists an invariant, vacuum-vector Ω = U Ω ∈ H.
• The quantum field ϕ is an operator-valued distribution.
• Vectors of the form ϕ(f1) · · · ϕ(fn)Ω, for f ∈ S and arbitrary n span H.
• The field ϕ transforms covariantly under U .
• The field ϕ is local.
• The space of invariant vectors Ω is one-dimensional.
Technically the Poincar´
e group G = {Λ, a} is the semidirect product of the Lorentz group (the
group of 4 × 4 matrices Λ that preserve the fundamental Minkowski quadratic form t2 − x2), with
the abelian group R4 of space-time translations x → x + a. The composition law for the group
is {Λ1, a1}{Λ2, a2} = {Λ1Λ2, Λ1a2 + a1}. Aside from the identity representation, the irreducible,
unitary representations of this group are infinite dimensional. On this account, one also requires
strong continuity of the unitary representation.
The strongly-continuous, unitary, irreducible, positive-energy representations of the Poincar´
e
group were dear to the heart of George Mackey. Many of these representations had been discovered
by the physicist Majorana in 1932 [38], and they were also worked on by Dirac and Proca. The
mathematical theory of the irreducible representations of the Poincar´
e group appeared in the ground-
breaking 1939 paper of Eugene Wigner [59]. This work formed a motivating example for Mackey’s
theory of induced representations.
If one considers the connected components of the Poincar´
e group (by ignoring reflections), these
representations are characterized in the case of space-time dimension d = 4 by two parameters
{m, s} called mass and spin. The mass m ≥ 0 is non-negative and the spin s is half-integer. The
representations of the group with reflections can be constructed from those of the group without
reflections, and both sorts of representations occur in the description of different particles in physics.
The self-adjoint, infinitesimal generators of the representations of various subgroups of the
Poincar´
e group play a special role in quantum physics. In particular, one writes the space-time
translation subgroup U (I, x) for x = (x , t) as
U (x) = eitH/ −ix·P / .
(III.1)
The self-adjoint operator H that generates time-translation is called the Hamiltonian, and it is
identified with the energy in physics. The self-adjoint generators P of spatial translations are called

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Quantum Theory and Relativity
7
the momentum. For simplicity, for the remainder of this article we use units in which Planck’s
constant
= 1, and also the velocity of light c = 1.
As the representation U is infinite dimensional, one expects that the Hamiltonian H is un-
bounded. A basic supposition of axiomatic field theory is that the energy has positive spectrum.
(One says that U is a positive energy representation.) Generally one adds a constant to adjust the
infimum of the spectrum to zero. The mass operator M is defined as the positive square root of
M 2 = H2 − P 2 .
(III.2)
The mass operator M commutes with the representation U . The vacuum Ω lies in the null space of
M , and one says the vacuum is unique if the null space is one-dimensional. If M has an eigenvalue
m > 0, the corresponding eigenspace is called a one-particle space in case U acts irreducibly on this
eigenspace.
In the case of a scalar field ϕ(f ) covariance means that the unitary representation U (Λ, a) of
the Poincar´
e group acts as
U (Λ, a) ϕ(f ) U (Λ, a)∗ = ϕ({Λ, a}f ) ,
where ({Λ, a}f ) (x) = f (Λ−1(x − a)) .
(III.3)
Furthermore for a massive, scalar field, the subspace ϕ(f )Ω ∈ H should lie in the representation
space of an irreducible representation with parameters m > 0 and s = 0, when the Fourier transform
of f is localized near a hyperboloid E2 −p 2 = m2 in energy-momentum space, where m is an isolated
point in the spectrum of the mass operator M .
The property of locality is the least intuitive of the axiomatic assumptions; it is a technical
way to embody the notion of relativistic causality. In quantum theory, one associates observable
quantities with self-adjoint linear operators on H (operators, for short). One says that the two
self-adjoint observables can be measured simultaneously if they can be simultaneously diagonalized,
namely if they commute. One also uses the term independent to describe commuting observables.
In case f ∈ C∞
, then the assumption of locality requires that ϕ(f )ϕ(g) = ϕ(g)ϕ(f ) when the
0 real
supports of f and g cannot be connected by a light ray. One says that such supports are space-like
separated. The locality condition for fermion fields ψ(f ) is somewhat more complicated, as these
fields are not assumed to be directly observable; rather quadratic functions of fermions are assumed
to be observables. This leads to a formulation of the locality criterion for fermion fields as the
anti-commutativity of the fermion fields, ψ(f )ψ(g) = −ψ(g)ψ(f ), if the supports of f and g are
space-like separated.
From these assumptions, one was able to conclude interesting consequences, both about the
analyticity of the expectations of products of fields, and also about symmetries of such field theories.
One of the best-known conclusions is a proof of a relation between the spin of a particle (integer or
half-integer) and its statistics (Bose-Einstein or Fermi-Dirac) respectively. The result is that the anti-
commutation of fields at space-like separation is incompatible with an integer-spin representation
for single particles, and likewise commutation of fields at space-like separation is incompatible with
a half-integer-spin representation for single particles. Another well-known conclusion is that nature
has a discrete symmetry called PCT, involving parity, charge conjugation, and time reversal.
Lehmann, Symanzik, Zimmermann, and Haag and Ruelle showed how to incorporate scattering
theory of (massive) particles into the axiomatic framework. They assumed more structure, especially
the existence of isolated, massive, one-particle states that one can obtain by applying a field to the
vacuum Ω. They then derived the existence of the scattering matrix for an arbitrary number of
particles, as well as formulas to compute the scattering matrix elements from the vacuum expectation
values of products of fields. Discussion and references can be found in [53, 36, 28, 20]. The additional
axiom of Haag and Ruelle enables a clean mathematical treatment of scattering for particles of mass
m > 0 (but it does not apply to the case of massless particles). The axiom corresponding to the
lightest particle having mass m says:

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Arthur Jaffe
• The mass operator M has isolated eigenvalues, 0 and m > 0, and a spectral (mass)-gap (0, m).
Haag later introduced a slightly more general axiomatic approach, emphasizing the algebraic
properties of the fields, see [28]. Rather than studying the representation U of G on H, he studied
the representation of the automorphism group that U implements on the algebra of field operators
A. In place of the vacuum vector, he studied the state (expectation) that it engenders, namely
a normalized, positive, linear functional on A. Yet another algebraic approach was introduced by
Borchers, see [3], focusing on the algebra of test functions f with which one pairs the fields.
These axiom schemes all appear simple and natural. However in themselves they embody very
few details about the physics; no laws of interaction are given! Thus one appears to have an enormous
freedom in the choice of examples. But for many years after the original study of this framework
in the 1960’s, there were only two sorts of examples that satisfy any of the axiom schemes. And
neither sets of these examples describe non-trivial interaction between particles, so they both are
lacking from the point of view of physics.
The first set of examples are the free fields, which satisfy linear equations and describe freely
moving (non-interacting) particles. The fields provided motivation for the formulation of the axioms.
While from the point of view of physics, freely moving particles are not extremely interesting, they do
provide a basis for discussion, and their existence shows that one can realize all the axiom schemes.
Furthermore if a set of interacting particles move away from one another, one expects that they
behave asymptotically at large times like free particles. Thus one expects that the free-particle
structure appears as a limiting case in the scattering theory of a field describing interaction.
The second set of examples were some exactly-soluble interactions in two-dimensional space
time. These exactly soluble models also did not lead to a robust description of particle interaction,
with scattering matrix elements often independent of the energy.
III.2
Examples with Interaction and Constructive QFT
However, free fields do not realize the hope that that one can describe the interaction particles
through an elementary equation of evolution for a relativistically covariant field. Motivated by the
lack of an interesting realization of axiomatic quantum field theory, a great deal of interest focused
on the question of whether specific non-linear quantum fields could be found. One attempted to
construct solutions to elementary non-linear equations that do not appear to be exactly integrable
or soluble in closed form.
Furthermore perturbation theory of the linear, free-field equation did not provide a good route
to find solutions. There was proof in some cases, and evidence in other cases, that the perturbation
expansions that one finds in all physics texts would not converge. And establishing a convergent
re-summation of perturbation series required a priori properties of a solution that one did not know
exists.
Clearly one must look for guidance from perturbation theory on how to approach the formulation
of the non-linear equations, and how to cancel their divergences. But one must also develop a new,
non-perturbative existence strategy, including a non-perturbative theory of renormalization. This
extensive investigation came to be known as constructive quantum field theory.
IV
No Interaction is No Problem: Free Fields
Before getting into the heart of how to treat interactions, let us first explore the structure of the
free-scalar field theory. If one considers freely-moving particles (corresponding to classical motion
without acceleration) then it is easy to find a Hilbert space H, a field ϕ, and a representation U
satisfying all the axiom schemes. The states of n particles arise from applying a polynomial of
degree n in the field to the vacuum state Ω. This field satisfies a linear Klein-Gordon wave equation.

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Quantum Theory and Relativity
9
We illustrate here the case m = 1, s = 0. In dimension d = 4 the space spanned by ϕ(f )Ω is the
representation space for a mass-1, spin-0 irreducible representation of the Poincar´
e group. A similar
construction holds for any d.
Consider for example a single particle moving freely in Minkowski space-time M d with coordi-
nates x = (x, t). In quantum theory one formulates the state h of a particle (the wave function)
as a function h(x) of the coordinate x ∈ Rd−1 at a given time. (We assume that space is d − 1-
dimensional, with d − 1 = 3 the standard case.) Compatibility with special relativity means that
the one-particle subspace K is the representation space of the spin-zero, mass-one representation of
(the Poincar´
e group) G.
Time evolution of the single-particle state h is given by the inverse U (t)∗ = e−itH of the one-
parameter time-translation subgroup of G.
This group is also called the Schr¨
odinger group of
quantum theory, and h(t) = U (t)∗h defines a solution to the Schr¨
odinger equation idh(t)/dt = Hh(t),
with initial data h.
IV.1
One Scalar Particle
We give here the details of the scalar case, so we take K to be the representation space of the scalar,
mass one-particle. This is the Sobolev space H− 1 (Rd) with the inner product
2
dp
h, h
= h, h
=
˜
h(p ) ˜
h(p )
.
(IV.1)
K
H
(Rd−1)
− 1
2ω(p )
2
1/2
Here ω(p ) = p 2 + 1
, and ˜
h = Fh denotes the Fourier transform1 of h,
1
(Fh)(p) = ˜
h(p ) =
h(x ) eip·x dx .
(IV.2)
(2π)d−1
Consider the space-time translation subgroup U (x) of (III.1). The natural choices of H and P
in Fourier space are
F−1HF = ω(p )
and F−1P F = p ,
(IV.3)
which entail
H = ω = (−∆d−1 + 1)1/2 ,
and P = −i
,
(IV.4)
where ∆d−1 denotes the Laplacian on Rd−1. Note
ω2 = P 2 + 1 .
(IV.5)
Thus the inner product can be expressed as
−
h, h
=
h, (2ω) 1 h
.
(IV.6)
H− 12
L2
Also the mass operator M has eigenvalue 1 on K = H− 1 . For h ∈ H− 1 , the function
2
2
f (x) = U (x , t)∗h ,
(IV.7)
is an H− 1 (Rd−1)-valued solution to the Klein-Gordon equation,
2
∂2
(
+ 1) f (x) = 0 ,
where
=
− ∆d−1 ,
(IV.8)
∂t2
1Note that throughout this paper we use the symbol ˜
f to denote the Fourier transform of f . One must be careful
to distinguish this from the symbol ˆ
f , which we reserve for later use: in §VII we define the quantization ˆ
f of f .

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Arthur Jaffe
with initial data,
f (x , 0) = U (x , 0)h ,
ft(x , 0) = −iU (x , 0)Hh .
(IV.9)
The function f (x) also satisfies the first order Schr¨
odinger equation
∂f (x , t)
i
= Hf (x , t) ,
(IV.10)
∂t
with initial data h(x ).
In the case of the s = 1 representation, the wave function h(x) is spinor-valued and satisfies
2
a first order differential equation (such equations were originally introduced by Dirac, Weyl, and
Pauli). In the case of spin s = 1 the wave function h(x) is vector valued, etc. We continue the
exposition of the scalar case.
IV.2
Fock Space of Many Particles
One can combine several particles without interaction by taking tensor powers of the Hilbert spaces
for the single particles. With identical particles, one uses a symmetric tensor product to combine
integer spin particles, or particles with integer spin and half-integer spin. One combines two particles
with half-integer spin using the anti-symmetric tensor product.
A simple way to describe an arbitrary number of non-interacting particles is by a Hilbert space
H that is the direct sum of the n-particle spaces Fn. Let F0 = C, and F1 = K, while for scalar
particles, the n-particle space is the symmetric tensor product
Fn = K ⊗s K ⊗s · · · ⊗s K = K⊗s n ,
(IV.11)
n factors
with the inner product normalized so that h⊗ n = h⊗s n has norm n!1/2 h n
. One takes H as the
H− 12
Fock space over H− 1 (Rd−1), which is the direct sum of n-particle spaces,
2
H = F = ⊕∞ F
n=0
n .
(IV.12)
The unitary U (x) acts on each tensor factor K in Hn, with n ≥ 1, and as the identity on H0. One
often denotes the function 1 ∈ F0 by Ω, and calls this no-particle state the vacuum in H. Also the
vacuum is translation invariant, U (x)Ω = Ω, so the state Ω has zero energy and momentum (as well
as zero angular momentum, etc.).
IV.3
Desired Properties for the Free Scalar Field
One desires the free scalar quantum field ϕ(x) to be an operator-valued distribution that also is a
solution to the Klein-Gordon equation
(
+ 1) ϕ(x) = 0 .
(IV.13)
With a real Schwartz-space function f ∈ S(Rd), one desires that ϕ(f ) =
ϕ(x) f (x) dx is a sym-
metric operator on F , transforming according to (III.3). The differential equation (IV.13) could also
be written as ϕ((
+ 1) f ) = 0.
The free field solution has the property that it is sufficient to pair ϕ with a test function on
Rd−1. Let δt denote the Dirac measure localized at time t. Then for h ∈ H− 1 (Rd−1), the operator
2
ϕ(h⊗δt) should be a continuous function of t on a dense domain. By choosing more regular functions
h, the field ϕ(h ⊗ δt) should become differentiable in t. Let
∂ϕ(h, t)
ϕt(h, t) =
= [iH , ϕ(h, t)] .
(IV.14)
∂t

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