Free Fields and Multilinear Algebra over Hilbert Spaces

over Hilbert Spaces

D . K A S T L E R

Institut de Physique Theorique, Marseille, Frame

The historical way in which free fields were introduced failed to reveal the simplicity of their mathematical structure. The construc­

tion of a free field as an «infinite set of harmonic oscillators » is indeed much more complicated than the definition obtained by means of multilinear algebra over the one-particle space. In present day field theory where free fields are considered as describing the asymptotic behaviour of coupled fields (rather than as representing their « zero-order approximation») it seems desirable to present free-field theory in its simplest mathematical context as one of the building stones of coupled field theory (rather than as the simplest example thereof).

In order to construct a free field we want: (a) the one-particle space H, which is the Hilbert space of classical normalizable solutions of the free field equation; (b) the corresponding Green function;

(e) the tensorial algebra &~(H) = @ H®0 0 ^p over H. More precisely we

3 >«o

need in the case of bosons the algebra £f(H) = φ SH®^P of symmetric

J>-0

tensors over Η and in the case of fermions the algebra &(H) =

= 0 AH®CO ^P of antisymmetric tensors over H.

P = 0

The construction (o) is independent of the special nature of the Hilbert space Η and can be done once for all for an arbitrary Hilbert space. Once this is done the problem reduces to points (a) and (b).

One obtains in this way a uniform description of free field which we shall apply to (I) the second quantized Schrodinger equation, ( I I ) the scalar neutral field, ( I I I ) the electron-positron field, (IV) the photon field.

Let us first describe the construction of ^(-ff), ^(B"), and for an arbitrary (complex) Hilbert space H. We denote by (f\g) the scalar product of the vectors / and g of Η (linear in g and antilinear in / ) . Let g(f) be a bounded antilinear form in fl", i.e., a (complex

D. KASTLER valued) function on Η such that *

φ(αή + α/') = « > ( / ) + α'>(/') and

(1) M / ) <

οι/ι

(/ and 0 are arbitrary vectors in JT, and a, a' arbitrary complex num*

bers, G denotes a non-negative constant and ||/|| =(/[/)). I t is a well known fact that to each such form φ there exists a vector g^veH such that one has for each feH

<p{f) = (f\9⁹)

the correspondence qxr-tgg, being one to one and linear. Η = Η®¹ appears in that way as (isomorphic to) the set of all bounded anti- linear forms on itself. We shall analogously define H®* as a subset of the set of all antilinear forms ^(/χ, A ,.···,/*) of ρ arguments f^t e H**

One could first think of defining the elements of H®^p as bounded antilinear forms, in the sense that for arbitrary A> /², · · -,fveH

(2) Μ/ι,/.Γ·^/»)Ι<σ||/¹||·|/.ΙΙ···ΙΙ/»1,

Where G is a constant. However this would not lead to a Hilbert space.*** In order to get a Hilbert space one has to resort to the fol­

lowing construction. Let g^lr g²,- · ·, g^P be arbitrary vectors of H, we define their tensor product g¹® g²®- · · ® g^p as the antilinear form

(flfi® j . ® · · · ® g⁹)(fi,U,---,fp) = ( / i li O ( / « I Λ ) · · · ( Α [ ί , ) , / . e . We next consider the set Λ®^ρ of all finite linear combinations of such tensor products of ρ vectors (these are again antilinear forms). Given two such linear combinations:

η

ψ = Σμ>9ΐ®9ί®···®9!,,

* W e denote hy α* the complex conjugate of the complex number a.

** <p(fi>

A> ···»

being kept fixed.

*** But rather to a Banach space.

* It is clear that the same antilinear form of h®^p can be written in an infinite number of ways as a linear combination of tensor products.

** This process is well known from the construction of real numbers starting from rational numbers. It consists in taking as the elements of H®^p the Cauchy sequences of elements of h®^p.

*** The condition (5) is of course much stronger than (2). It reduces to (1) for p = l . For p = 2 (2) would imply that <p(f¹⁹ /²) be the matrix elements (A 1^ I A) °f ^a bounded antilinear operator, whilst (5) implies for Β a condition of the Schmidt-Hilbert type, which is stronger than complete continuity.

their scalar product shall be defined as

Η m

{4)

(φιy) =

Σ

• · ·

I</',),

which can be shown to pertain to the antilinear forms φ and ψ and not to the special way in which they are defined in (3) as linear com­

binations of tensor products.* (<p\y>) is obviously linear in ψ and anti- linear in <p. The set h®^p with the scalar product (4) is then an incom­

plete Hilbert space which we turn into a (complete) Hilbert space H®^p by the standard process of completion.** The elements of H®^p will be called p-tensors on H. They can be shown to be antilinear forms

<p(hiUr" Ί A ) o n -H" ^{s u c}h that *^{o r} arbitrary vectors f'^peH^} 1 < h < p, l < i < η

(5) ι i

λ

(η,

t- 1 » - l

where G is a constant. ***

Notice that if u is an element of R and φ an element of If®*

the antilinear forms

ψ'(A, A , · · · , = (A I u) φ(A , . / » , · · · , A + i ) ,

and

<?"(A, A r · · , A - i ) = Α , · · · , A - i )

belong, respectively to H®¹*¹ and H®*-¹ (9/' is only defined for ρ > 2).

We give it a meaning for ρ = 1 by deciding that H®° be the one- dimensional Hilbert space of complex numbers and for ρ = 0 by setting it equal to zero). The transformations φ -+φ' and φ ->φ" are linear mappings from ΖΓ®^Ρ, respectively, in H®¹*¹ and Η®»-¹, which we de­

note, respectively, by P{u} (product by ueH) and G{u) (contraction by UEH).

Β. KASTLER

We call a tensor symmetric if it is invariant on permutation of its arguments, antisymmetric if it is multiplied by the signature of the permutation. The linear subspace of symmetric and antisymmetric p-tensors are obtained by applying in H®^p the corresponding ortho­

gonal projectors

σεΟν

Ρ * σεθρ

where the σ are the elements of the permutation group G^P of the first ρ integers (with signatures χ(σ)) acting in 3®^v according to

, U im- , f * ) = <pUoirfo%r-, ίο») j fiEH,

8 is the symmetrizing operator, A the antisymmetrizing operator (S² = S and A² = Α). Ιί Η denotes a one-particle space the states of the system of ^-particles are described by the elements of SH®^P for bosons and by those of AH®* for fermions.

In order to define the Hilbert space of field theory (free or coupled) we now consider the direct sum

3>=0

the elements of which are infinite sequences of vectors <p^kGH®^v,

Φ = (<pu) = (ψο, ψυ" ·, φ*,-·) ·

&~(H) is a Hilbert space with the following definitions of linear com­

bination and scalar products):

αΦ + α'Φ' = (oc(p^k + α'φί)

for Φ=(φ^Λ) and Φ'=(φ^Λ) (Φ\Φ') = Σ(φ*\φί)

k-l

(the elements of &~(H) are such that ( Φ | Φ ) < oo). The definition of of the operators P{u}, G{u} and A is extended to f(H) by writing Ρ{η}Φ = Φ{η}(^Ψο, ^Ψι,-· ·, ^ψΛ,-. ·) = (0, P{u)(p,^rP{u}(p^u-· ·, P{u}<p^k-^W>-) 0{η}Φ = C{u}(<p^0l<p^lr-· ·, <ρ*,· · -) = (0{η}φ^1} C{u}(p^2r-· ·, 0{η}φ^Μ9' · ·) 8Φ = ( % , . % , . · · · , S<p^k,^:--)

ΑΦ = (JLy^e, .

I t is easily seen that P{u} and G{u} are hermitically conjugate, 8 and A are hermitian and 8² = S, A² = A. 8 and A project (orthogonally) in &(Η) on the respective subspaces

y(H) = 8 F(H) = φ 8H®*,

J>«=0

and

9(H) = A F(H) = © ,ΙΗ®*,

Ρ =0

which are built up, respectively, with the sequences (y^k) of sym­

metric and antisymmetric tensors. (On H®°, 8 and A are defined as equal to the identity. £f stands for symmetric, 9 for Grassmann).

For a boson or a fermion with one-particle space Η the space of field functional will, respectively, beSf(H) or 9(H), whereby SH®'=AH®» =

= H®° represents the vacuum state.

We now go over to the definition of creation and annihilation operators. Given any one-particle state, ueH, its creator a⁺{u}, and annihilator a~{u} in S?(H) will be defined as

(6)

(6a)

a+{u} = y/NSP{u}8 a-{u} = SO{u}Sy/ff, its creator b+{u} and annihilator b~{u} in 9(H) as

b+{u} = <^K/NAP{u}A b-{u} = AC{u}A^/W, whereby y/W is the operator on J~(H) given by

^ΝΦ = y/E(<p^or(pi,- Ί <P*i" ') = (°> <Pu Λ/2^²,· · ·, v ^ * ' * · ·) . One can consider the α^±{ ^ } as operating on £f(H), the 5±{w} as ope­

rating on 9(H). They satisfy on these spaces the familiar commu­

tation (anticommutation) relations:

[a±{u}, ±a{u}]- = 0 ί [b±{u}, b±{v}]⁺ = 0 [a-{u}⁷ a+{v}]„ = (u\v)

j [&-({«},

(7)

a~{w}; b⁺{u}, b~{u} are pairs of hermitically conjugate operators.

The creators a+{u}, b+{u} depend linearly on ueH *:

a+{<xu + α V } = oca+{u} + <x'a+{u'}

b+{<xu + ol'u'} = <xb⁺{u) + 0L'b+{u'} .

* The annihilators depend antilinearly on u.

D. KASTLEtt

One easily shows that for any (normalizable) ue J2*, b^u} is a bounded and a^u} an unbounded operator. W e finally want to mention an important «covariance property» of the creation and annihilation operators: let U be a unitary or antiunitary transformation of the one-particle space 21, we denote again by U the transformation ® U®^p

f»=0

induced in &~(H) (it can be considered as well as operating on Sf(H) or on &(H)). One then has the relations

j ^a± { ^ } = Ua^ujU-¹

( 8 ) j b±{v^u} = U±b{u}U-\

which are useful in the proof of Lorentz invariance.

Let us now apply the previous notions to the description of the free field mentioned above.

I. Second Quantized Schrodinger Equation

(a) The one-particle state Ξ is the Hilbert space of normalized solutions ψ(ξ, τ ) of the Schrodinger equation

( 9 )

* ^ = { - ^ + ^ ^ > >

the norm deriving from the scalar product (independent of τ ) , (γ(ξ, τ ) I?'(ξ, r)) =f γ>*(ξ, τ)γ>(ξ, τ ) d? .

(b) The Green function is the Feynman propagator of (9), i.e., the solution 1£(ξ, τ ; χ, t) of (9) in the variables (ξ, τ ) such that

#(?,<;*, t)=d£-x).

If the γ><(ξ, r) form a complete system of orthonormal solutions of (9) one has

(10) ^ ( ξ , τ ; * , ί ) = | ν ί( * , ί )^ν( ξ , τ ) .

1 - 1

i f is furthermore such that for any ψ (ξ, τ) e IT * (11) ( ^ ξ , τ ; * , * ) [ γ > ( ξ , τ ) )

* The scalar product in (11) being effected with respect to the variables ξ, τ

(o) The state of space functional is ^(JET). The field operator Ψ(χ, t) is given by

Ψ(χ, t)=b-{K&r;x, *)},

where _Κ(ξ, τ ; χ, t) is considered as a (formal) element of Η depending on the parameters x, t. As Κ satisfies in the variables *, t the conju­

gate of Eq. (9), Ψ(χ, t) obeys the «field equation » (9) in *, t. Ac­

cording to (7) and (11) one has the anticommutation rule [ψ(χ,

*),

oi

=*(*,*;*', η.

As ϋΓ(ξ, τ ; Λ, ί) belongs only formally to Η, Ψ(χ, t) is not a bona fide operator on Η but rather an operator-valued distribution. In order to get a properly defined operator one has to «smear it out» with a test function /(#), i.e., to calculate

f(x)y(x, t)dx

II. Scalar Neutral Field

(a) The one-particle state Η is the Hilbert space of normalizable positive energy solutions φ(ξ) of the Klein-Gordon equation *

( • - m²M ? ) = 0, the scalar product being defined as

<->

(9(hW'M) =φ*{ξ) ^φ'(ϊ)*%·

t

The Lorentz transformation H induces in Π the transformation y C E -¹? ) if Ls&

^ ( J C -¹! ) if I e J ? * .

This transformation is unitary or antiunitary according to LeJ?* or LeSeK I t wiU be called V(L).

(b) The Green function is equal to ιΔ⁺(ξ). I t is such that

(12) U(L) [i Δ+] = i Δ+, LeS?

Ψ(ξ)

* Symbols like f, χ denote four-vectors in Minkowski space.

Β. KASTLER

III. Electron-Positron Field

Let us denote the contravariant spinors (the (numerical) vectors of spin 4-space) by symbols like ψ, the corresponding covariant spinors being ip (the antilinear mapping ψ -> ψ of spin-space into its dual space is usually written ψ ψ). We denote furthermore by (yT, ψ) the invariant scalar product on spin space (usually written ψψ') and call C the conjugation of spin space commuting with Dirac's γ operators and with the transformations (L) of spin space induced by the Lorentz transformations L.

(a) The one-particle space Η is the direct sum and such that for any <ρ(ξ)βΗ

(13) (ιΔ+(ξ-χ)\φ(ξ)) = φ(χ).

(c) The space of field functionals is £f(E). The field operator A(x) is given by

A(x) = a+{i Δ+(ξ — χ)} + a~{i Δ+(ξ

—

where ιΔ⁺(ξ— af) is considered as a (formal) element of JET depending on the parameter x. A(x) evidently satisfies the Klein-Gordon equation in x. As a consequence of (7) and (13) one has the commutation relation

[A(x), A ( # ' ) ] - = ί(Δ⁺ (χ—χ') — Δ+(χ'—χ)) =ΙΔ(χ — χ') . Owing to the fact that ιΔ⁺(ξ— χ) is a distribution which belongs only formally to H, A(x) is an operator-valued distribution. One gets a bona fide (unbounded) operator on S?(H) by smearing out A(x) with a test function /(#), i.e., taking

jf(x)A(x) d*x.

The Lorentz invariance of the theory, due to the « covarianee » of A(x), is an immediate consequence of (8) and (12). For any Lorentz trans­

formation U(L) one has the relation

A(Lx) = U(L)A(x)U(L)~¹.

of two isomorphic Hilbert spaces corresponding, respectively, to elec­

trons and positrons. J3"^el is the Hilbert space of normalizable posi­

tive energy solutions ψ(ξ) of the Dirac equation (γ^μ ψ + ™)

Ψ(1)

with the scalar product

«Kf) I Ψ'(£)) =f (V(f), ϊ⁴Ψ(ξ)) <*ξ .

The Lorentz transformation L induces in H^el the transformation [called U(L)]

ψ ( ξ } ~* I G(L) $(L~^), LeSeK

is isomorphic to H^el and will be distinguished from it by writing with a dash the variable of the positron wave functions ψ(ξ').

(b) The Green function is the distribution — ί8'(ξ) acting linearly on spin space. Writing 8^±(ξ) = (γ^μΰ^μ — ηήΔ^) and denoting by a * the adjunction of spin-space operators with respect to the metric (ψ, ψ), one has the properties

(15) 8^±(Lx) =

CS±(x) = — S*(W)C, (16) ^(ξ-χ)^\ψ(ξ))= (*,ψ(χ)) .

(ο) The space of field functionals is &(H). The contravariant field operator ψ(β, χ) (depending linearly on the covariant spinor s) is defined as

y(£, x) = b-{iS+tf- x)s} + b+{iCS-(f-x)s} ; the corresponding covariant operator is

y)(s, x) = {y>(£, x)}* .

The contravariant and covariant coordinates ψ"(χ) and ψ^α(χ) are ob­

tained by taking dual bases e^a and ε* in spin space and its dual space and writing

= ®) 9

Υ . KASTLEU

One has according to (7) and (16) the anticommutation relations

[γ(£, £')]+ = - #(# — Χ')?') , 8 = 8+ = 8-,

and according to (8) and (15) the covariance properties, f((L)s, Lx) = U(L)y)(s, x)V(L)-^x, y>(C(L)7, Lx) = U(L) φ(7, x)U(L)~^l.

IV. Photon Field

(a) Here one has to consider three different one-particle spaces which we call K, L, and I . Κ is defined as the space of normalizable positive energy solutions φ(ξ) of the D'Alembert equation

(17) ^ΠΦΒ) = 0 , with the scalar product

(18) {Φ{Ξ)\Ψ(Ξ)) = ί [ ( ν ( ϊ ) , ^ 9 ' ( ί ) ) ύ ξ , t

which defines an indefinite metric owing to the signature of the Min­

kowski product standing under the integral sign. (18) can be written as [ ( f / ( E) l£ ( U ) d£>(I),

where Ω is the invariant measure on the positive light cone [dfi(fc) =

= (dfc/2|fc|)] and 9>,(Λ) the invariant Fourier-transform of φ(ξ):

φ(ξ) = (2n)-*^q>,(k) exp [i(k, χ)]άΩβ) .

In addition to (17) we impose on our wave functions the so-called Lorentz condition

(19) a^ ( f ) = ο

equivalent to

(19a) (k, ^Vf(k)) = 0 .

This specifies a linear subspace L of Κ on which the metric defined by (18) is positive but not positive definite: choosing for each k a Lorentz frame such that k¹ = k2 = 0, k* = k° = | k | the Lorentz con-

dition indeed reduces to

<p*(k) = φ°(1) (φ^Το), (p²(k) arbitrary);

the norm of φ(ξ) e Ζ is therefore equal to

J{ | ^ W l² + k²W l²} c l i 2 W ,

which shows that the vectors of L with vanishing norm are of the form

ψ

(1) =

They therefore build a linear subspace L⁰ of L orthogonal to L. This shows that we get a bona fide Hilbert space (with positive-definite metric) by taking the set L = L/L⁰ of all classes of elements of L modulo an arbitrary element of L⁰. The Lorentz transformation L induces in Κ the transformation V(L) defined by *

j Σψ(Σ-^) if Le&

φ®~*\-Σφ*(Σ-*Έ) if Lz&K

U(L) is unitary or antiunitary with respect to the scalar product (18) according as ϋ ε ^ or It leaves the space L⁰ invariant and can easily be shown to act on Σ (to transform classes modulo L⁰ in classes modulo I⁰) .

{b) The Green function is ιΔ⁺(ξ) as for the neutral scalar field.

It can be considered as acting multiplicatively on photon spin space.

(c) The theory of the photon field makes use of the spaces Sf(K), S?(L) and ^ ( X ) . The elements of Sf(L) with vanishing norm build a linear subspace spanned by the sums of tensor products con­

taining at least one vector of vanishing norm. Sf(L) can be obtained as the set of classes of elements of Sf(L) modulo an arbitrary element of S?⁰: y(L) =^(L)/9'⁰. The indefinite metric of Κ induces in Sf(K) the indefinite metric of Gupta and Bleuler.

The field operator A(~s, x) (depending linearly on the spin vector J) is defined on ^(K) as

A(s, x) = A+(s, x) + A~(s, χ), A±(89 x) = a±{i — x)l} .

* There should be no confusion between the subspace L of Κ and the Lorentz transformation L denoted by the same symbol.

D. KASTLER

Its components are obtained by choosing a basis β^μ in Minkowski space and writing Α^μ(χ) = Α{~β^μ, χ). A(s, χ) does not leave the sub- space Sf(L) invariant, whence the necessity of considering the spaces Κ and^(Jf). y(L) can be shown to be the set of 0eSf{K) such that

d^fiA~fⁱ(x)0 = 0

for all x. Gauge invariant operators like Ε^μν(χ) can be shown to be defined on £?(L). Their matrix element containings states of are equal to zero.

The group defined by the infinitesimal operators (gauge-group), F^x =j{d^MA»M - λΜ^μΑ»} άχ ,

t

for all λ satisfying (17) and (19) determines in S?(L) the classes mo­

dulo £f⁰ which build up the space Sf(L).

EEFERENCES

J. M. Cook, Trans. Am. Math. Soc, 74, 223 (1953).

V. Fock, Z. Physik, 75, 622 (1932).

D. Kastler, Ann. Univ. Sarav Ciencia Scientia, 4, 206; 5, 86 and 204 (1956).

«Introduction a Telectrodynamique quantique», Dunod, Paris, in press.

K. Potier, J. Phys. Badium, 18, 422 (1957).

J. Gr. Valatin, J. Phys. Badium, 12, 131 (1951).

A. S. Wightman and S. S. Schweber, Phys. Bev., 98, 812 (1955).