Application of distributions to the theory of elementary particles in quantum mechanics

Elements of the Theory of Distributions Let Rn denote the n-dimensional Euclidean space and let flRn or simply fl be the space of all complex valued functions qJ defined in Rn which ha

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J Willard Gibbs could, after ten years of thought, summarize his ideas

on a subject in a few monumental papers or in a classic treatise His competition did not intimidate him into a muddled correspondence with his favorite editor, nor did it occur to his colleagues that their own progress was retarded by his leisurely publication schedule

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v

ELLIOTT MONTROLL GEORGE H VINEYARD MAURICE LEVY

is a revised version of the Berkeley lectures notes of 1960 Therefore

it is not written and planned as a textbook Each part was written after the lecture had been given-hence some repetitions or even modifi-cations in the book This text contains a large amount of pure mathe-matics: theory of vector valued distributions, tensor products of topological vector spaces, Fourier transforms and Bochner theorem, Hilbert sub-spaces and associated kernels have been exposed as such, but many proofs have been omitted, being intended primarily for physicists I tried to make a distinction between scalar valued functions

or distributions printed in normal characters, and vector valued, printed

in bold characters But this tedious distinction has been progressively abandoned throughout, and everything is written in normal letters, even for vectors, except in special cases where the distinction is important

L SCHWARTZ

V11

Scalar and Vector Particles in an Arbitrary Universe 10

2 THE SET~ OF THE UNIVERSAL PARTICLES AND ITS STRUCTURE 15

4 ELEMENTARY PARTICLES AND THE LORENTZ lNVARIANCE 68

IX

X CONTENTS

Supports of Extremal Measures

Mesons

Lorentz Invariant Distributions

Determination of all Mesons

Description of :¥f for the Meson

5 SPIN PARTICLES

Vector Particles

Determination of all Vector Particles

Complete Description of~ :¥f for Vector Particles

The Electron

Vector Particles with Zero Mass

6 DEFINITION OF SOME PHYSICAL NOTIONS

Scalar Case

Vector Case

The Intrinsic Parity

APPENDIX: Density of Probability of Presence of

of the position of a particle being in a region Ac:R3 at any time t is JJJil/J(x, y, z, t}l2 dxdydz Note f/1 must be square integrable for each t

A

and we assume ~J itft(x, y, z, t)jl dx dy dz = 1 for each t If we define an inner product f!i t/t1(x, y, z, t)lfl 2 (x, y, z, t) dx dy dz, then the function

f/1 belongs to a Hilbert space for each t

In non-relativistic quantum mechanics, f/1 satisfies the Schroedinger wave equation:

where m is the mass of the particle and ~ is the Laplacian

In relativistic quantum mechanics, space and time are not separate; thus one cannot say that f/1 is a function of four variables, unless a Lorentz coordinate system is chosen In order to treat space and time together, the space E4 , a four-dimensional affine space, is introduced and f/1 is defined on E4 An affine space will be defined later

1

2 STUDY OF ELEMENTARY PARTICLES

Definition A particle, Jf, is a Hilbert space of functions on E4

Definition A motion, f/1, is an element of :¥f with Ill/Ill = 1

Let a be an arbitrary Lorentz transformation in E4 and G be the Lorentz group Under the transformation a, a function f/1 goes into a function af/1

Definition If, for all a E G,

f/1 E Yf => af/1 E Yf

II a l/IIIJ( = lll/lll:re then the particle, Yf, is a universal particle In short, a universal particle

is a particle that does not change under a Lorentz transformation

Definition A universal particle, Yf, is elementary if :¥f contains no subspace which transforms into itself under all aeG, i.e., :¥f is minimal

We shall show later that the space :¥f depends on a parameter

m 0 ~ 0 and a parameter taking on the two values + and - The +

parameter is interpreted as the charge and m 0 as the rest mass of the particle

Definition A meson is a scalar elementary particle (i.e., the wave function f/1 is a scalar)

For a system of two particles, the Hilbert space has the same axioms

as before, except that its elements are functions on E4 x E4 Only systems of one free particle will be dealt with in these lectures

For the sake of generality, we shall assume that our Hilbert space

is not a space of functions but a space of distributions

We, therefore, begin with a short introduction to the theory of distributions

Elements of the Theory of Distributions

Let Rn denote the n-dimensional Euclidean space and let fl)(Rn)

(or simply fl)) be the space of all complex valued functions qJ defined

in Rn which have derivatives of all orders and which vanish identically outside a bounded region in Rn The functions qJ will be called testing functions Note that fl)(Rn) is a linear space

We introduce now a topology in fl)

Definition A sequence of testing functions { qJj(x)} converges to zero in fl) if all the functions qJ j(x) vanish identically outside the same bounded region in Rn and if the functions qJ i(x) and all their derivatives converge uniformly to zero

POSITION OF THE PROBLEM 3

Definition A distribution T is a continuous linear functional on ~' i.e., the image under T of an element qJ E 9} is a complex number denoted

by < T, 4>) such that

(T, (c1(/J1 + C2({J2)) = c1 (T, (/J1) + c2(T, ({J2)

and

({Ji -; 0 implies that < T, ({Ji) + 0

Let ~'(Rn) (or simply ~') denote the space of distributions on Rn

Example Letfbe a locally integrable function in Rn

Then

<f, qJ) = J f(x)qJ(x) dx = J f(x}qJ(x) dx

defines a distribution Here A is a bounded region in Rn (the support

of qJ ) Thus every locally integrable function defines a distribution Clearly, /1 and /2 define the same distribution if and only if /1 = /2

almost everywhere Considering the Lebesgue classes defined by this relation (i.e., identifying functions which are equal almost everywhere),

we conclude that the Lebesgue classes of locally integrable functions form a subspace of the space of distributions

Other important examples are the Dirac distribution, ~, defined by

( ~' cp) = ({J(O)

or

(~(a), qJ) = qJ (a) and the dipole ' defined by

(,, qJ) = - qJ'(O)

Definition The derivative of a distribution Tis defined by the formula:

(T', ({J) = -(T, qJ') From this it follows that

(T<m>, ({J) = ( -l)m (T, (/J(m))

( -; , ({J)= -(T,-;-)

4 STUDY OF ELEMENTARY PARTICLES

where p denotes the n-tuple of integers p = {pb , Pn),

IPI = P1 + · · · + Pn

and

DP = (~)Pt (~)P2 (~)Pn

Thus every distribution has derivatives of all orders

Example Consider the Heaviside function Y(x) defined by

Y(x) = { ~ Then

00

x>O x<O

-oo

= - J qJ'(x) dx = qJ (0) = < £5, qJ)

0 Therefore Y' = £5

Definition Letfbe a continuous function and let A = {x :f(x} =f= 0} The closure A of A is called the support of the function f

Definition Let n be an open set in Rn and let T E ~' We say that

T = 0 in Q if < T, qJ) = 0 for all qJ E ~ whose support is contained

in Q For example, £5 = 0 in R - {0}

Theorem Let {!li} be any system of open subsets in Rn and suppose

that T = 0 in every Qi Then T = 0 in u!li

Proof We must show that < T, qJ) = 0 for every qJ E ~whose support

is contained in u!li Let A be the support of some qJ E ~ Since A

is compact and covered by {!li}, there exists a finite subcover {nik},

k = 1, , n Let { t/Jk}, k = 1, , n, be an infinitely many times

con-tinuously differentiable partition of unity on A with respect to Qik'

that is, t/J k E ~ (Rn), each t/J k has its support in Q ik and

n

L t/Jk = 1 k=l

on A Then

(T, (/)) = (T, L t/Jk(/J) = L (T, t/Jk(/J) = 0

Corollary For every distribution T there exists exactly one maximal

open subset of Rn in which T is zero

POSITION OF THE PROBLEM 5

Proof Consider all Qi in which T = 0 Then u Qi is the required set

Definition The support of T is the complement of the maximal open

subset of Rn in which T = 0

We introduce now a topology in the space of distribution~' Since

it is a linear space it suffices to define convergence to zero

Definition Weak convergence: Let {1}} be a sequence in~' We say that 1j converges to zero in the sense of distributions, or 1j + 0 in

~',if (1j, ({J) + 0 for every qJ E ~

Strong convergence requires a certain uniformity and it will be defined when needed

Theorem Differentiation is a continuous operation, i.e 1j + 0

in p)' implies that 7j' + 0 in ~'

Proof (Tj, cp) = - (1j, cp') + 0 for every cp E ~

Remarks The weak topology defined here makes convergent a lot

of sequences which are ordinarily divergent A series which is gent in the sense of distributions may be differentiated term by term, i.e., if T = L1J then T' = LT)

conver-Theorem Let jj + 0 almost everywhere and suppose that l!JI ~ g,

where g is a fixed positive locally integrable function Then jj + 0 in the sense of distributions

Proof This follows from the Lebesgue convergence theorem

Example A trigonometric series

L ak e2nikx

k

is convergent in the sense of distributions if and only if lakl ~ Aka, for

k =F 0, where A is a constant and et is some positive integer Thus many trigonometric series become convergent in the sense of distri-butions To see this consider the series

6 STUDY OF ELEMENTARY PARTICLES

Examples The series

Differentiating term by term we see that

00

L (2nik) e2nikx

-oo converges to

00

L l5'(x - k)

k=-00

£5' £5' £5' £5' £5' -2 -1 0 1 2

Affine Spaces: Lorentz Transformations

In the previous section we defined the space ~'(Rn) of distributions

on the Euclidean space Rn In a similar way we may define the space

~'(En) of distributions on then-dimensional vector space En However,

in physical space there is no pre-determined origin, so that we do not have an En to start with For this reason, we introduce the concept of an affine space

Definition An affine space is a set E and an associated vector space E

This association is defined by a map from E x E to E which maps a pair a, b of elements of E to the vector a b of E, and such that the following two laws are satisfied :

(1) Chasles' relation: If a, b, care any three elements of E, then

(2) Let o be a fixed element of E The map a + oa is a one-to-one

correspondence between E and E

It should be noted that (1) may be generalized to more than three

POSITION OF THE PROBLEM 7 elements Furthermore, according to (1) the triple a,a,a yields 3 a a = 0

or a a = 0 and the triple a,a,b yields a b + ba = 0

For obvious reasons, the notation

ab = b- d

is very convenient Thus the difference between two elements a,b of E

is a map which maps the pair a,b to the vector a b of E, and which obviously satisfies the above two laws If a is a given element of E

and xis a given element of E, then there exists one and only one element

bEE such that a + x = b where this equality is equivalent to

(Jd- (Ja = a(b -a)

Note that the associated linear operator a is uniquely determined

by (J Furthermore, the composition of two affine operators is an affine operator and the invertible affine operators form a group

Example The translation U : x -+ x + U is an affine operator from the affine space E onto itself The associated linear operator of a

translation is the identity operator,

Let eb e2, , en be an orthonormal basis in E, i.e., (eilei) = 0 fori =f j and (eilei) = + 1 Every finite dimensional vector space with a non-degenerate quadratic form has an infinite number of orthonormal bases However, the number of basis elements e such that (ele) = + 1

8 STUDY OF ELEMENTARY PARTICLES

and the number of basis elements e such that (ele) = -1 is independent

of the particular chosen basis

Definition The signature of an n-dimensional vector space with respect to a given quadratic form (xI y) is the pair of integers {p, q), where p + q = n, p is the number of O.N basis elements e such that (eje) = + 1 and q is the number of O.N basis elements e such that (eje) = -1

Definition A Lorentz four-dimensional vector space is a vector space with a quadratic form which has the signature (3,1) The orthonormal basis will be denoted by eb e2, e3, e0, where (eilei) = + 1, i = 1,2,3 and ( e0 I e0 } = -1 A Lorentz four-dimensional affine space is an affine space E4 whose associated vector space E4 has the signature (3,1)

By a Galilean reference system we mean a chosen origin 0 in E4 and a chosen orthonormal basis eb e2, e3, e0 in E4 (chosen coordinate system)

Every point of the universe has four coordinates x1, x 2, x3, x 0 = ct,

three of space and one of time

Definition A Lorentz transformation a is an affine invertible operator

in a Lorentz affine space which preserves its Lorentz structure, i.e., the associated linear operator preserves the quadratic form

(axlav) = (xlv)

The Lorentz transformations form a group The group G consisting of all the Lorentz transformations (J will be called the inhomogeneous Lorentz group, whereas the group G consisting of the associated linear operators a will be called the homogeneous Lorentz group

Example Translations are Lorentz transformations

One may now define the space ~'(E) of distributions over the affine space E More generally, one can define the space ~' for a manifold V,

as the space of infinitely differentiable functions with compact support, with a suitable topology, and ~'(V), space of distributions on V, as its dual

Universal Scalar Particles

Now the definitions of a scalar particle and a universal particle will

be made more precise

Definition A scalar particle in the universe E4 is a set :¥f satisfying the postulates :

(1) :¥f is a vector subspace of ~'(E 4 )

POSITION OF THE PROBLEM 9

(2) :¥f is equipped with a Hilbertian structure, that is, there is a linear antilinear form ( t/1 11 t/1 2}Jf' (linear in t/1 1 and antilinear in t/12) in Je

which is Hermitian and positive definite, and :Yf is complete with respect to the norm II t/1 IIJf' = (t/11 t/J)}e

(3) The canonical embedding of :¥f into ~' is continuous, that is,

t/J i + 0 in :¥f => t/J i + 0 in~'

We shall find that :¥f represents charged particles If the distributions

in E4 were restricted to be real valued, then :¥f would describe a neutral particle

Definition A motion of a particle is an element t/1 E :if such that

II t/1 II Jf' = 1

A universal particle (universal with respect to the Lorentz group) is one which is considered the same by different observers An observer makes his observations in some frame of reference; thus the particle :¥f

is interpreted by him as being a space of distributions over R4 instead

of E4 If all observers interpret j'f to be the same space of distributions over R4 then :¥f is a universal particle A more precise definition is given after the operation of (J E G on distributions is defined

A Lorentz transformation (J E G not only operates on E4 but also

on every structure given over E4 If <p(x) for x E E4 is a complex function on E4 , the transformation <p + (JqJ is defined by the equation

(J<p( (JX) = <p(x)

or, equivalently,

From the fact that a E G is a linear operator, it follows that :

Theorem <p E ~(E4} => u<p E ~(E 4 } It follows from the definition of

(JqJ that:

Theorem <fJn + 0 => (J(/Jn + 0

Thus (J gives an automorphism of~ onto ~

The operation of (J on distributions is defined by the equation

10 STUDY OF ELEMENTARY PARTICLES

It is simple to show that the operation of a followed by -r on ~' is the same as the operation of -ra on ~' Then it follows that a is an automorphism of~' onto ~'

Given an affine space E and a positive measure on E which is invariant under translation, a measure on E is uniquely defined

Then any locally integrable function! onE defines a distribution

(f, qJ) = Jf(x) qJ(x) dx

Given a quadratic form on E, there corresponds orthonormal bases

and a Haar measure In view of the fact that any a E G preserves the quadratic form, it will also preserve the Haar measures It follows that

a preserves the correspondence between functions and distributions for this measure

Given a scalar particle :¥f c~'(E 4 ) and any a E G, one may form

the space a Jf, the set of at/J for all t/1 E Jf With the inner product

( at/J 1l at/J 2) u £ = ( t/1 1l t/1 2)£

the space a :¥f is also a Hilbert space

Definition A scalar particle :¥f is universal if for all a E G the following

is true:

(1) a:¥( = :¥(

(2) II at/J IIJt' = II t/1 IIJt' for all t/1 E :¥f ·

It follows that :¥f is a universal particle if and only if every a E G is a unitary operator of :¥f onto Jf

Scalar and Vector Particles in an Arbitrary Universe

Definition A universe V is a C 00

-manifold of finite dimension n A group G whose elements operate on V will be called the structure group of the universe

POSITION OF THE PROBLEM 11 Definition A scalar particle in the universe V is a set :¥f satisfying the postulates:

(1) :¥f is a vector subspace of @'(V), the space of distributions in V

(2) :¥f is equipped with a Hilbertian structure

(3) t/J i + 0 in :¥f => t/J i + 0 in @'(V)

Definition A scalar particle :¥f in the universe V is universal (with respect to G) if for all (J E G :

(1) (J :¥( = :¥(

(2) II (Jt/1 IIJt' = II t/1 IIJt' for all t/1 E :¥f ·

Example For one scalar particle, we may take V = E4 with a given Lorentz quadratic form and the corresponding Lorentz group as the structure group

Example For two particles, we take V = E4 x E4 The structure group G is again the Lorentz group acting on E4 x E4 as follows: For (x, y) E E4 x E4 and (J E G

(x, y) + (J(X, y) =((JX, (Jy)

In order to treat particles such as the electron, proton, etc., we must introduce the concept of a vector-valued distribution Let F be a finite dimensional vector space over C

Definition An F-valued distribution T on V is a continuous linear map T: qJ + (T, qJ) of @(V) into F

The space .@'(V; F) of F-valued distributions on V, the space c!t'(.@(V); F) of continuous linear maps of @(V) into F, and the tensor product ~'~V) ® F of @'(V) and F are all identical:

.@'(V; F) = c!t'(.@(V); F) = .@'(V) ® F

Example Let V = Rn be an affine space with a Lebesgue measure

If f(x) is a locally integrable F-valued function on Rn, then to f ponds a distribution

corres-qJ + (f, qJ) = J f(x) qJ(x) dx

If S E .@'(V) and f E F, then the vector-valued distribution Sf E

.@'(V; F) may be defined by the equation

(Sf, ({J) = (S, ({J) f

Sf is identified with S ® f E .@'(V) ® F

If F has the basis

12 STUDY OF ELEMENTARY PARTICLES

then T e ~'(V; F) can be written

a e G + -r E the set of unimodular operators in F, which form a group, such that an infinite number of elements of G correspond to each element of this group of linear operators in F, and the action of each a

on any element of F is the same as the action of the corresponding -r given by the mapping

Definition The operation of a on T E .@'(V) ® F is defined by the equation

a( ( T, qJ)) = (aT, a <P),

or, equivalently,

(aT, 1/1) = a( (T, a-1 t/1)) = -r( (T ,_ t/1( a 0x)))

Definition A universal F-valued particle in the universe Vis defined

in exactly the same way as a universal scalar particle in the universe

v

POSITION OF THE PROBLEM 13

Weak and Strong Convergence

Definition Let E be a topological vector space A set A c: E is called convex if whenever x, yEA the elements ax + (1 - a)y, 0 ~a~ 1, also belong to A E is called locally convex if its topology can be defined

by a base consisting of convex sets

Let E be a locally convex topological vector space and let E' be its dual, i.e., the space of continuous linear forms on E We shall define weak and strong convergence in E'

Definition The sequence { ej} c E' converges weakly to zero, ej + 0 weakly, if (ej, e) + 0 for every e E E Here the inner product is the one defined naturally as being equal to the value of ej at e

Strong convergence requires a certain uniformity on the bounded subsets of E

Definition A subset A of E is called bounded if it can be mapped into any neighborhood of zero by a contraction with a non-zero ratio For example, if E is a Banach space, a subset of E is bounded if it can

be mapped into any ball by a contraction with a non-zero ratio

Definition A sequence { ej} c E' converges strongly to zero, written

ej + 0 strongly, if < ej, e) + 0 for every e E E and this convergence is uniform on every bounded subset of E

Let us return now to the spaces @(V) and @'(V) @(V) is the space

of testing functions defined on the universe V If K is a compact subset

of V, let @x(V) denote the space of testing functions whose support

is inK In @x(V) we may introduce the norms

II (/J lim = sup IDPqJ(x)l

xeK

IPI~ m

where DP denotes the {pb p 2 , , Pk) derivative Just as before, we define

convergence to zero of a sequence {({Jn} in @x(V) by requiring that

II (/Jn II m + 0 for all m

An element T of @'(V) is a linear form on @(V) which is continuous

on every @x(V) A sequence {1j} c @'(V) converges weakly to zero,

1j + 0 weakly, if < 1), qJ) + 0 for every qJ e @(V) It converges strongly

to zero, 1j + 0 strongly, if < 1), qJ) + 0 for every qJ E .@(V), and this convergence is uniform on the bounded subsets of.@ x(V) for any K

We state here without proof the following important theorem:

Theorem The space .@'(V; F) of F-valued distributions on V is a

14 STUDY OF ELEMENTARY PARTICLES

locally convex topological vector space which is complete under the strong topology

From this point on, our basic purpose is to find all the subspaces

:¥f of ~'(V; F), such that :¥f may be equipped with a Hilbertian structure and convergence in :¥f implies convergence in ~'(V; F)

{ Jf, (I)Jt} consisting of a linear subspace :¥f of E and a scalar product

on :¥f satisfying the following conditions :

(a) Provided with (I)Jt, :¥f is an Hilbert space;

(b) The injection of :¥f into E is continuous; that is, convergence in :¥f

implies convergence in E

On ~ we may define the following:

(1) Multiplication by non-negative scalars, A ~ 0; the set

AJf = { :¥f if A > 0

OifA=O

If Te :¥f and therefore TE AJf, then

(2) Addition: The set

:¥f 1 + :¥f 2 = {T: T = T1 + T 2 , T1 E :¥f b T 2 E :¥f 2 }

with the norm

(3) Order: A partial ordering is defined in ~ by the relation

:¥f 1 ~ :¥f 2 if :¥f 1 c :¥f 2 and the norm in :¥f 1 is greater than or equal

16 STUDY OF ELEMENTARY PARTICLES

dual space E' into E, that is, E /::- E', L is continuous and L(.Ae') =

"XL(e') L is called positive if (e', Le') ~ 0 for all e' E E' The duality

product between E and E' is defined by

(e',f) = e'(f), the value of e' atf, for feE, e' E E'

Example Let E = en, the n-dimensional complex vector space, then

E' = en An anti-kernel here is a positive definite hermitian matrix L,

L1 ~ L 2 if (e', L1e') ~ (e', L 2 e') for all e' E E'

We prove now the following fundamental result:

Theorem There is a one-to-one correspondence between the elements

:¥f of~ and the positive anti-kernels L To :¥f E ~ corresponds the

kernel L = J i Jt t J, where J is the natural injection :¥f + E, tJ its

transposed E' + Yf' and iJt the canonical anti-isomorphism Jf' + Jf

Proof First we show that to a given :¥f corresponds a positive anti-kernel L Let e' E E' Since e' is a continuous linear functional on E and the injection of :¥f into E is continuous, it follows that e' is a continuous linear functional on Jf By the Riesz representation theorem there is a unique element of :¥f which we shall denote by Le', such that

(e', h) = (hi Le').JF, hE Jf, e' E E' (2.1)

Clearly the map L: E' + :¥f c E which is defined by equation (2.1} is

anti-linear To show that L is continuous, let ej + 0 in E' (strong topology), i.e., (ej, h) + 0 for every hE E and uniformly for h on

bounded subsets of E Since the injection of :¥f into E is continuous,

it follows that bounded subsets of :¥fare bounded in E Hence (ej, h) =

(hI Lej) + 0 uniformly on the unit ball of Jf Therefore II Lej II Jt + 0

SET~ OF THE UNIVERSAL PARTICLES 17

and L is continuous Finally, if we let h = Lf' in equation (2.1) we obtain

(e', Lf') = (Lf'ILe').JF, e' f' E E',

and if e' = f', then

(e', Le') = (Le'ILe')Jf' ~ 0, e' E E'

which shows that L is positive

(2.2}

Conversely, we must show that to a given positive anti-kernel L

corresponds an element :¥f of~ Let :¥f 0 = L(E') c Jf According to equation (2.2), we define in :¥f 0 the inner product

(ulv)Jf'o = (Le'ILf').JFo = (f', Le'), u, v E :¥f 0 (2.3) where u = Le', v = Lf' We prove now for :¥f 0 the following:

(a) Definition (2.3) of (u I v)Jf'o is unique, i.e., independent of the choice

of e' and f' such that u = Le' and v = Lf' This follows immediately

from equation (2.3} and the hermitian symmetry by noting that

(u I v) .Jt>o = 0 if either u or v is zero

(b) The form (u I v) .Jt>o is positive definite Since L is positive,

(ulu)Jf'o = (e', Le') ~ 0, u E :¥f 0

If (e', Le') = 0, then by Schwarz's inequality,

l<f', Le')l ~ <f', Lf')! (e', Le')! = 0 for allf' E E' Hence Le' = u = 0 and (ulu)Jf'o = 0 if and only if u = 0 (c) The topology of .Yf 0 is finer than that of E, i.e., the injection of

Yf 0 into E is continuous It suffices here to show that the unit ball of

Yf 0 , { Le': (e', Le') ~ 1 }, is a bounded subset of E From Schwarz's inequality

I <f', Le')l ~ <f', Lf')! (e', Le')! ~ <f', Lf')!

it follows that the set { <f', Le') : (e', Le') ~ 1, e' E E} is bounded in (f

for each f' E E', or that the unit ball of :¥f 0 is weakly bounded in E

Using now Mackey's theorem, which states that a subset of a locally convex topological vector space is strongly bounded if and only if it is weakly bounded, it follows that the unit ball of :¥f 0 is bounded in E

We have shown up to now that :¥f 0 = LE' is a pre-Hilbert space

whose injection into E is continuous We expect to obtain the required

18 STUDY OF ELEMENTARY PARTICLES

:¥f corresponding to L by completing :¥f 0 It is necessary therefore to prove the following:

(1) If there exists an :¥f corresponding to L such that equation (2.1)

is satisfied, then :¥f 0 is dense in Jf Let hE :¥f such that (h!Le').tt = 0

for all e' E E' Then (h!Le').tt = (e', h) = 0 for all e' E E' and, by Hahn-Banach theorem, h = 0 Thus an element of :¥f which is ortho-gonal to every element of :¥f 0 is zero and hence :¥f 0 is dense in Jf

(2} The completion :it 0 of :¥f 0 can be embedded in E Consider the (continuous) injection

and its unique extension to

J

:¥to-~ E

.*'0 J E(= E)

We must show that J is still an injection LethE.*' 0 and let lz = Jh E E

We claim that for every e' E E'

If hE :¥f 0 , then h = h = L]' and equation (2.4) simply reduces to the definition (2.3) of the scalar product in :¥f 0 Consider now the sequence {hv} c :¥f 0 such that hv + h Then

(e', hv) = (hv!Le').JF 0

Passing to the limit and using the continuity of the scalar product in

:it 0 and the continuity of the forme' we obtain equation (2.4) Suppose now that h = Jh = 0 Then, by equation (2.4), (h!Le')£ 0 = 0 for all

Le' E :¥f 0 , and since :¥f 0 is dense in :it 0 it follows h = 0 Thus J is one-to-one

(3) Finally let :¥f = J :it 0 and transfer the Hilbert structure of :it 0

to Jf We must show that :¥f is associated with L If k E Jf, then there

is an hE :it 0 such that ih = h = k and equation (2.4) yields

(e', k) = (e', h) = (h!Le')~0 = (k!Le').tt

which shows that L is the anti-kernel associated with Jf

We shall give now another construction of the Hilbert space :¥f

corresponding to a given positive anti-kernel L This construction will

be from above, in contrast to the one given in the proof of the previous theorem, which was from below

Theorem Let L be a positive anti-kernel An element hE E belongs

SET~ OF THE UNIVERSAL PARTICLES 19

to the Hilbert space :¥f corresponding to L if and only if

e' E E' (e', Le')

and if this condition is satisfied, then

llhll Jt = e' sup E E' (e', Le')t· l<e',h)l

Proof If hE Jf, then, using equation (2.1) and Schwarz's inequality, l<e',h)l = l(h!Le')Jtl ~ (hlh)~ (Le'!Le')~ = llhii:K (e',Le')t for all e' E E'

Conversely, suppose that equation (2.4) holds and consider the map

Le' + (e', h)

It may be easily verified that this map is an anti-linear functional on

Yf 0 which is continuous, since it is bounded on the unit ball of :¥f 0

Therefore, it may be continued to a continuous anti-linear functional

on the completion :¥f of Yf 0 Hence: by the Riesz representation theorem there exists k E :¥f such that

Le' + (e', h) = (k!Le')Jt = <:', k)

for all e' E E' By Hahn-Banach theorem, h = k E Jf

The Structures of tl and !!' + (E', E)

Let !l'(E', E) denote the set of continuous linear maps from E'

into E, !l'(E', E) denote the set of continuous anti-linear maps, or anti-kernels, from E' into E and !!' +(E~ E) denote the set of positive anti-kernels It was shown in the previous section that there is a one-to-one correspondence between ~ and !!' +(E', E), ~ ~ !!' +(E', E)

It was also mentioned that we may define a natural structure (addition, scalar multiplication, etc.) in both ~ and !!' +(E', E) In this section the structures of~ and !!' +(E', E) will be defined more precisely and the correspondence between them will be established

Definition In 2 + (E', E) we define the following:

(1) Order relation: L1 ~ L 2 if L 2 - L1 ~ 0 i.e if L 2 - L1 is a positive anti-kernel

(2) Multiplication by a non-negative scalar A.: (A.L) (e') = L(A.e')

20 STUDY OF ELEMENTARY PARTICLES

(3) Addition: (L1 + L2} e' = L1 e' + L 2 e'

Correspondingly in ~ we define the following:

( 1 ') Order relation: :¥f 1 ~ :¥f 2 if :¥f 1 c :¥f 2 and the norm in :¥f 1

is greater than or equal to the norm in :¥f 2, II h II .}f'

1 ;;:::: II h II Jf'2•

(2') Multiplication by a non-negative scalar A:

To :¥f corresponds the space

A:¥f ={{0} if A= 0

:¥f if A > 0, and the norm

(3') Addition: To (Jf b :¥f 2} conresponds the space

Jf1 + :¥f2 = {h:hEE,h = h1 + h2,h1 E:¥f1,h2E:¥f2}

and the norm

It should be noted that (1) and (1') indeed define order relations since we have :

(a) :¥f 1 ~ :¥f 2 and :¥f 2 ~ :¥f 1 imply that :¥f 1 = :¥f 2

(b) L1 ~ L 2 and L 2 ~ L1 imply that L1 = L 2

The first follows trivially from the fact that, in a Hilbert space, the scalar product is uniquely determined by the norm The hypothesis

of the second means, by definition, that

(e', (L2 - L1} e') ;;:::: 0, (e', (L1 - L2 ) e') ;;:::: 0, e' E E',

or

<e' L e')=(e' L e') e' E E' ' 1 ' 2 ' • Using now the formula

4 (e', Lf') = (e' + f', L(e' + f')) - (e' - f', L(e' - f'))

+ i (e' + if', L(e' + if')) - i (e' - if', L(e' - if'))

we conclude that

(e', L1f') = (e', L 2 f'), for every e',f' eE'

SET~ OF THE UNIVERSAL PARTICLES 21 and by Hahn-Banach theorem it follows that

LJ' = LJ'', f' E E'

It should also be mentioned, as a consequence of the requirement

II h II.JF1 ;;:::: II h II Jt 2 in the definition of :¥f i ~ :¥f 2, that convergence in

.Yf i implies convergence in :¥f 2 •

We shall prove now the correspondence between the structures of

~and !!' +(E', E)

Theorem (1) If :¥f i and :¥f 2 correspond to Li and L 2 , respectively, then :¥f i ~ :¥f 2 if and only if Li ~ L 2 •

(2) If :¥f corresponds to L, then A.Jf corresponds to A.L(A ;;:::: 0)

(3) If :¥f i and :¥f 2 correspond to Li and L 2 , respectively, then :¥f i + :¥f 2

(e', Lie')~ (e', L 2 e')

hence :¥f i c :¥f 2 and, if hE Yt' b II h II Jt1 ;;:::: II h II.JF2·

(2) This follows immediately from the choice of the norm in A.Jf:

1 1 < e', h > 1 1 < e', h > 1

llhllu = .JA sup (e',Le')t =sup (e',A.Le')t

(3) We must first say what we mean by the Hilbert space :¥f i + :¥f 2 •

22 STUDY OF ELEMENTARY PARTICLES

The set :¥f i + :¥f 2 consists of all elements h of E which may be written

in the form h = hi + h2 with hiE :¥f i and h2 E :¥f 2 If :¥fin :¥f 2 = {0} the norm in :¥f i + :¥f 2 may be defined by

II h II;, 1 + 82 = II hi II;, 1 + II h II;, 2

where h = hi + h2 is the unique representation of h If, however,

:¥fin :¥f 2 =/= {0}, then an element hE :¥f i + :¥f 2 has an infinite number of representations since the zero element has an infinite number

of representations of the form a - a, where a E :¥f i n :¥f 2• For this reason we take as the definition of the norm

II h 11;, 1 +82 = inf( II hill;, 1 + II h2ll;,2) where the infimum is taken over all possible representations

h = hi + h2 It must be shown that this is actually a norm, that it may

be defined by a scalar product in :¥f i + :¥f 2, that :¥f i + :¥f 2 is complete and the topology of :¥f i + :¥f 2 is finer than the topology of E Since this procedure seems rather tedious, we shall construct the Hilbert space :¥f i + :¥f 2 in a different way

Let :¥f i ~ :¥f 2 denote the abstract Hilbertian (direct) sum of :¥f i

and :¥f 2 An element of :¥f i ~ :¥f 2 is a pair (hb h2) where hiE :¥f i and

h 2 E :¥f 2 • Addition and scalar multiplication are defined by

(hb h2) + (kb k2) = (hi + ki, h2 + k2),

A(hb h2) = (Ahi, Ah2),

and the scalar product in :¥f i ~ :¥f 2 is defined by

((hi, h2)1(ki, k2))81ff>82 = (hilki)81 + (h21k2)82 Note that :¥f i ~ :¥f i ~ {0} and :¥f 2 ~ {0} ~ :¥f 2• Let us map the abstract Hilbert space :¥f i ~ :¥f 2 into E by

SET~ OF THE UNIVERSAL PARTICLES 23

The map Jf i ffi Jf 2/% + E is an injection whose image is the space

.Yf i + Jf 2 We can transfer the structure of Jf i ffi Jf 2/.% on Jf + Jf 2;

it is seen immediately that the factor norm on Jf i ffi Jf 2/.% is~ the infimum norm defined above in Jf i + Jf 2 By this construction, the space Jf i + Yf 2 is already proved to be a Hilbert space contained

in E, whose topology is finer than that of E

We must still show that Yf i + Yf 2 corresponds to Li + L 2• Let

L = Li + L 2 and let :tt denote the Hilbert space corre~ponding to L

We must show that :Yt = Jf i + Jf 2 Since L ;;:::: Li and L ~ L 2

it follows that :Yt => Jf i and :Yt => Jf 2 ; hence :Yt => Jf i + Jf 2 Furthermore,

:Yt o = {Le':e' eE'}

is contained in Jf i + Jf 2 since Le' = Lie' + L 2 e' Thus we have

:Yt => Jt'i + Jf2 => :Yto

where :Yt 0 is a dense subset of :Yt Since both :Yt and Jf i + .Yf 2 are complete- Hilbert spaces it suffices to show that the norm of :Yt 0

is equal to the norm of Jf i + Jf 2 Let h = Le' = Lie' + L 2 e' = hi+ h 2 , where hi= Lie'eJt'i and h 2 = L 2 e'ee1l' 2 • We have

(e', Le') = (e', Lie') + (e', L 2 e')

or

llhll2% 0= llhill~i + llh2ll~2;;:::: llhii~1+Jf2

In order that equality holds it is sufficient to show that the element

(hb h2} of Jf i ffi Jf 2 is orthogonal to the null space %, because in this case there is no reduction in norm If (ni, n2} E.% we have

(hilni)81 + (h2ln2).re2 = (Lie'lni)81 + (L2e'ln2)82

= (e', ni) + (e', n2) = (e', ni + n2) = (e', 0) = 0

Remark If Jf i ~ Jf, then there exists one and only one Hilbert space Jf 2 such that Jf i + Jf 2 = Jf This follows from the fact that

L 2 = L - Lb is a positive anti-kernel

Definition The anti-kernels Li and L 2 are called disjoint if the only anti-kernel which is less than or equal to both Li and L 2 is zero, i.e.,

if L ~ Li and L ~ L 2 implies L = 0

Definition The convergence of a sequence of anti-kernels

{Li} c !!' +(E', E) to an anti-kernel Lis called

24 STUDY OF ELEMENTARY PARTICLES

(1) point-wise convergence if Lie' + Le' for all e' E E'

(2) bounded convergence if Lie' + Le' for all e' E E' and this convergence

is uniform on bounded subsets of E'

It should be noted that !l' +(E', E) is a closed subset of ff'(E', E)

under both of the above convergences Furthermore it is a convex cone

It is a cone with vertex the origin since tL is a positive anti-kernel for

t ;;:::: 0 provided that Lis positive, and it is convex since tL1 + (1 - t) L 2

is a positive anti-kernel for every 0 ~ t ~ 1 provided that L1 and L 2

are positive We may therefore state the following

Theorem There is a one-to-one correspondence between the space

~ and a closed convex cone in a topological vector space This cone does not contain any ver;tor subspace other than {0} The last sentence

of this statement expresses the fact that the relation :¥f 1 ~ :¥f 2 is indeed an order relation

We turn now to the following question Given a system {OJ of elements of E, what are the necessary and sufficient conditions in order that it forms a complete orthonormal system for a Hilbert space

:¥f E ~? We observe first that an element e E E defines the following maps:

(a) a linear functional onE':

f' + (f', e), (b) an anti-linear functional onE':

f'-+ (f', e), (c) a hermitian form onE' x E':

f'eE',

f'eE'

(f', g') + (f', e) (g', e) (d) a map from E' into E, denoted by ee:

f'~ (f',e)e

It can be easily shown that ee is a positive anti-kernel The sponding Hilbert space :¥f = { Ae: A E C} with the norm II Ae II = IAI

corre-and the scalar product (AeiJ-Le) = Ap_

Theorem In order that a given set { eJ iei of elements of E be a Hilbert basis, i.e., a complete orthonormal system of a Hilbert spaceJf E ~,

it is necessary and sufficient that:

(1) The series of anti-kernels L eiei is pointwise convergent or that the

iel

finite partial sums of the series are pointwise bounded;

SET~ OF THE UNIVERSAL PARTICLES 25

(2) The set { ei} iei is Hilbert -free, i.e if { cJ iei is any set of complex numbers such that L lcil2

< oo and if Lei ei = 0, then ci = 0 for all

iel iel

be the corresponding anti-kernel Then, for every f' E E',

Lei ei f' = Lei <f', ei> = L elLf'lei) = Lf'

so that L eiei converges pointwise

iel

Furthermore, since { ei} iei is a Hilbert basis, it follows immediately

that it is Hilbert-free

Conversely, assuming that (1) and (2) hold, let us show that { ei}iei

is a Hilbert basis of some :¥f E ~ We first notice that condition (1)

means that for every f' E E' the partial sums of L (ebf') ei are strongly

iel

bounded in E, and for every f' and g' in E', the partial sums of

L (ei,f') (eb g') are bounded Taking/' = g' and using an elementary

such that L lxil2

< oo, and let l~ denote the subset of 1 2 whose

Let {xi}iei E l~ such that L lxil2 ~ 1 Then, for every f' E E',

iel I<Lxiei,f'>l = ILxi (ei,f'>l ~

~ (Lixil2)t(LI<ei,f'>l2)t ~

~ (LI<ei,f'>l2

)t < oo

26 STUDY OF ELEMENTARY PARTICLES

Let us extend now the above map to the completion of l~, which can

be identified with 1 2 :

by

Using the Hilbert-free hypothesis (2) it may be shown that this map is

an injection Let :¥f be the image of this injection with structure the transferred structure from 1 2 • The set { ei} iei is the image of the canonical basis of 1 2 and hence it is a Hilbert basis for Jf

Corollary If L is a positive anti-kernel, then L has an infinity of decompositions of the form

Scalar Particles

Let us return now to the study of scalar particles in a universe V

Recall that a universe is a C00

-manifold of dimension n and a scalar

particle is a Hilbert space :¥f with continuous injection in the space @'(V) of distributions on the universe V The locally convex topological vector space @'(V) is the dual of the space @(V) of infinitely differenti-

able functions with compact support on V An element T of @'(V) is

a distribution whose value on an element qJ of @(V) is denoted by

(T, qJ); .@(V) is a reflexive space, i.e., the dual of @'(V) is @(V) Each

of the spaces @(V) and @'(V) has the strong dual topology of the other

We shall apply now the results of the last two sections forE = @'(V) and E' = @(V)

In order to find the Hilbert spaces in @'(V) we must look for the anti-kernels L: @(V) + @'(V) We shall start by looking for the continuous linear maps from @(V) into @'(W) where V and W are two C00

-manifolds, for example, two Euclidean spaces Let y denote a

generic point of V and x denote a generic point of W For convenience,

we may denote .@(V) by @Y and @'(W) by .@~ Let !'}'(W x V) or @'x, Y denote the space of distributions on the product W x V (distri-butions of two variables)

Theorem (Theorem of kernels) The topological vector space !!' b

(.@(V); @'(W)) of continuous li~ear maps from ~(V) into @'(W) with the topology of bounded convergence is canonically isomorphic with

the topological vector space @'(W x V)

SET~ OF THE UNIVERSAL PARTICLES 27 Let K be a given element of .@'(W x V) K defines a continuous linear map v + K ·v from @(V) into @'(W) by the formula

(K ·v, w) = (K, w ® v), wE @(W) (2.5) where w ® v = w(x)·v(y) We must verify first that K·ve.@'(W)

Clearly K ·vis a linear functional in w If w + 0 in @(W}, then w ® v + 0

in ~(W x V), and, since K E .@'(W x V), (K ·v, w) = (K, w ® v) + 0

We must also show that v + K ·v is a continuous linear map The linearity is obvious If v + 0 in @(V}, then it is easily seen that (K · v, w) + 0 for every wE @(W), and this convergence is uniform when w

remains bounded in @(W)

For more clarity we shall repeat this result in terms of the alternate notation @ Y' .@'X' @ x, Y' and we shall give an example from the theory of integral equations from which this notation was originated Let

Kx,yE @'x,y be a given distribution in the two variables x andy Kx,y

defines a continuous linear map from @ Y into @~:

v(y) + Kx, y · v(y) = (K · v)x, v(y) E.@ Y'

where (K · v)(x) E.@~ is defined by the formula

be the value of a distribution T at the testing function qJ K(x, y) defines

a continuous linear map from .@Y into.@~:

28 STUDY OF ELEMENTARY PARTICLES

continuous linear map from @(V) into @'(W) defines a unique bution on W x V The proof of this is considerably more difficult, and

distri-it will be omdistri-itted Instead, we shall tum now to the proof of the equivalence of topologies of the isomorphic spaces !!'(.@ Y' .@~) and .@~, y· There is a natural topology in both these spaces In .@~, Y we shall take, as previously, the strong topology (uniform convergence on bounded subsets of ~x,y)· In !l'(.@y, .@~) we liave point-wise and

bounded convergence Let !!' b(.@ Y' .@~) denote the topological vector space consisting of the vector space !£'(.@ Y' .@~) with the topology of bounded convergence A sequence in !l'b(~Y' @~)of continuous linear maps from .@Y into .@~ converges to zero in the sense of bounded convergence if the image sequence of every element in .@Y converges

to zero in .@~, and this convergence is uniform on bounded subsets of

~y· We shall prove only that convergence in @'x,yimplies convergence

in !!' b(.@ Y' .@~) Let { K i} be a sequence in .@~, Y such that K i + 0 strongly, i.e., < K i' <p(x, y)) + 0 for every <p(x, y) E .@ x, Y' and this convergence is uniform on bounded subsets of .@x,y· We must show that for a fixed

v(y) E .@Y, (Ki·v, w) + 0 for every w(x) E ~x' and this convergence is uniform when w(x) runs over bounded subsets of .@x· By definition,

(Ki·v, w) = (Kj(x, y), w(x)v(y)) + 0

and the last convergence is uniform when w(x) remains bounded, since

in this case, w(x)v(y) remains bounded Again, the proof of the converse will be omitted

Remark The Banach-Steinhaus theorem implies that any weakly convergent sequence in.@' is also strongly convergent

Now that we have identified the space!!' b(.@(V); .@'(V)) of continuous linear maps from @(V) into @'(V) with the space ~'(V x V) of distri-butions of two variables, /it is a simple matter to obtain the space

!!' +(.@(V); @'(V)) of positive anti-kernels from @(V) to @'(V) and the space ~ of Hilbert spaces :¥f with continuous injection in @'(V) A distribution of two variables Kx, Y E ~'(V x V) defines a continuous linear map from ~(V) into @'(V):

SET~ OF THE UNIVERSAL PARTICLES 29

The positivity of this anti-kernel is defined naturally as follows:

Definition The anti-kernel v + K ·v defined by the distribution

Kx, Y E .@'(V x V) is called positive if

(Kx,y' qJ{X)@ qJ{y)) ~ 0

for every qJ E .@(V)

According to our general theorem, the relation between the positive

£-anti-kernel L and the corresponding Hilbert space :¥f is

T = K · t/J in the above formula, we obtain

(K·t/J, ({J) = (K, qJ ® l/i) = (K·t/JIK·qJ) (2.7) for every qJ and t/J in @(V)

Remarks In formula (2.6) no error should be made by putting a bar over T The space Yf is not invariant under conjugation The space

Yf = {T: T E Yf} represents the same particle as :¥f but with opposite charge

The definition of positivity of the anti-kernel defined by Kx, Y

originated from the theory of integral equations Let K(x, y) be a continuous function on V x V The kernel K(x, y) is called positive if for every set of elements {x1, x 2, ••• , x1} in the space V and every set

of complex numbers { z 1, z 2 , ••• , z1} the following inequality holds:

l

L K(xi, xi)zizi ~ 0

i, j

It may be shown that the definition of positivity of the anti-kernel

defined by Kx, Y E .@'(V x V) coincides with the above definition when

Kx, Y is a continuous function K(x, y) of the two variables x andy

We summarize the final results of this section in the following theorem:

Theorem Let V be a C00

-manifold The space~ of the Hilbert spaces

Yf with continuous injection in @'(V) in canonically isomorphic to the

30 STUDY OF ELEMENTARY PARTICLES

subspace of .@'(V x V) consisting of distributions Kx,y of two variables such that

(K, qJ ® qJ) ~ 0 for every qJ E @(V) The relation between such a K and the corres-ponding Jf is given by the formula

(T, ({J) = (TIK·qJ}Jf', TeJf, qJE.@(V), where the distribution K · qJ E Jf c.@'(V) is defined by

(K qJ, t/1 > = (K, t/1 ® qJ ), t/1 E .@(V)

Tensor Products

In order to generalize the results of the previous section to vector particles we must introduce the concept of a tensor product We shall present only the basic definition and properties without any proofs

Let E and F be two vector spaces For our purpose, it is not necessary

to give a complete definition of the tensor product The tensor product

of E and F is a new vector space E ® F with a given canonical bi-linear map from E x F into E ® F:

exf +e®f

E ® F is not the image of E x F under this map However, E ® F is

generated by elements of the form e ® f, i.e., every element x of E ® F may be written as

x = e 1 ®f 1 + e 2 ®f 2 + + ek®fk

The image under the canonical bi-linear map is not a vector subspace

of E ® F, but E ® F is formed by finite sums of elements of the form

e ®f We state now the following properties

If E and F have finite dimensions m and n, respectively, then the

dimension of E ® F is m · n

If { ei} and {fi} are bases of E and F, respectively, then { ei ® fi} is

a basis of E ® F, i.e., every element x of E ® F may be written in a unique way as

X = " l J X· l) ·8 · l 15(\ \C;I f · J

i, j

IfF is finite-dimensional with {!;} as its basis, then every element

SET~ OF THE UNIVERSAL PARTICLES

x e E ® F may be written in a unique way as

X = L 'j ® fj, 'j E E

j

31

If G is any vector space over the field of scalars C then G ~ G ® C

The isomorphism is g + g ® 1 since { 1} is a basis of the vector space

C over C

Suppose now that E and F are topological vector spaces We want

to define a topology in E ® F In general, there are several distinct such topologies However, if F is finite-dimensional, then there is a unique topology defined on E ® F, the topology of coordinate-wise convergence Let {fj}, j = 1, , n be a basis of F Every element x e

E ® F can be written as

n

X = L 'j ® fj, 'j E E

j= 1

A sequence { xk} in E ® F converges to zero, xk It 0, if'' It 0 for all j

This topology is independent of the basis of F Every "good" property

of E is also possessed by E ®F If E is locally convex, reflexive, or complete, then E ® F is also locally convex, reflexive, or complete, respectively

Ket G be a given vector space If p: E x F + G is a given bi-linear map, then there is a unique linear map u: E ® F + G such that

p(x~ y) = u(x ® y)

for every (x, y) e E x F Conversely, if u: E ® F + G is a given linear map~ then there is a unique bi-linear map p: E x F + G such that the above relation holds :

ExF

/~

E®F +G

u

If u is given, then p is immediately defined by the above formula If p

is given, the above formula defines u on elements of the form x ® y, and, since every member of E ® F is a finite linear combination of such elements, u is defmed on E ®F This important result demon-strates that the main use of the tensor product is the linearization of bi-linear maps: every bi-linear map p: E x F + G may be replaced