normal form expansion in an algebraic way

Pram[.na, Vol. 18, No. 6, June 1982, pp, 525-532. ~) Printed in India.

Normal form expansion in an algebraic way

p

MARTIN RIVAS Departamento de F/sica, Facultad de Ciencias, Universidad del Pa/s Vasco Aptdo. 644, Bilbao (Spain)

MS received 22 February 1982

Abstract. It is shown that in an algebraic approach, normal form expansion of operators, bounded or not, but densely defined on Fock space, converge weakly.

Keywords. Fock space; creation operators; annihilation operators; particle number representation; weak convergence.

1. Introduction

In many textbooks (Bogolubov and Shirkov 1959; Schweber 1964; Itzykson and Zuber 1980) little attention is paid to the normal form expansion of operators and usually it reduces to a recipe in order to make the vacuum state a zero-energy state or simply to be an eigenvector with a zero eigenvalue for general observables. Berezin (1966) shows that any bounded operator is representable in normal form and the expansion converges strongly. Chamorro (1978) shows that any operator (bounded or not) defined on the dense linear manifold formed by finite linear combinations of the basis in the occupation number representation, admits a unique formal expansion in terms of creation and annihilation operators. Following this paper, we show in the present work that this formal expansion indeed converges weakly to the operator.

In § 3 we review the method for the explicit calculation of the coefficients in the bosonic case, generalizing this result to the fermionic case (§ 5) and to a general system of rn kinds of bosons and n kinds of fermions (§ 6). In § 4 weak convergence is proved. Finally § 7 is devoted to some discussions about the vector space structure of the algebra of operators.

2. Preliminary notation

Let us first introduce the following notation: Let B be the set of sequences formed by non-negative integers, such that all of them but a finite number are zero; if

a ~ {al, ~2 .... ~t,...} is one of such sequences, i--1 at < Go. Let ~ =.[~1, a2,...

~ .... ]- and/3 = {/31,/3z,.../3, .... ]- be two such sequences; we can define another sequence 7 = a q- t , such that 7~ = ~ -t-/31, endowing B with a semigroup structure. In general a -- ~ ~ B, but we can define on B a partial ordering by saying ~/>/3 i f ~ - - ~ E B .

525 P.--5

526 Martin Rivas

Given any a there exists a finite number

GO

N . = n i = 1

of sequences which are not greater than a. Let B × B be the set of pairs of sequences. We can .~end to it the semigroup

and partial order structures by:

and (a, fl) f> (/~, v) if (a,/3) -- (/~, v) E B x B. Given any (a,/3) there exists a finite nut," cr of pairs which are not greater than

it. This number is Na N#. Within them, the number of pairs ~ , v) which verify that a - - /z = /3 -- v is

cA)

N ~ = lI (min (a,,/3,) q- 1}. i = l

We shall denote by in]m, m = I .... , k a collection of sequences which are less or equal to a given a.

Given any a E B, we associate to it two non-negative integers, namely

lal= i----1

(t)

a ! = II ai!, (2) i=1

where 22 and II are convergent since there exists an index n, such that for i > n a i = 0 , a n d 0 ! = l .

3. Bosonie systems

Let H B be the symmetric Fock space of a system of identical bosons. In the occu.

pation number representation, there is a one-to-one correspondence between the elements of B and the set of orthonormalised basis vectors of H B. Given a E/3,

we define the basis vector [ ~ ) = I al, a3,'", ai .... )" They verify

(~ I/3~ = 8~, (3)

1 i f a = /3 where 8,/3 = 0 otherwise.

If in this representation one has chosen an orthogonal set of one-particle states {f~}, i = 1,..., ~ , each a, represents the number of particles in the state f~.

Normal form expansion in an algebraic way 527

Let a* and at be respectively the creation and annihilation operators (CO and AO for short) for a particle in the statej~. They verify the canonical commutation relations,

[a,, a~] = [at*, a*l = 0,

[a,, a) ~ ] = S u, (4)

and are defined on the dense linear domain D, the linear span of the basis {] a > / a EB}.

Given any a E B, we define two operators on D:

oO

= n (a*) (a*) "x . . . . i---1

oo

= II (aL) a t = (a l ) a l . . . . ( a , )a ' . . . , (5) i=1

where for avoiding problems of convergence we agree to consider the right hand sides of (5) as consisting only of finite product operators for those ai 4 0. Then ~* and "~ are respectively finite products of CO's and AO's.

By acting on the vacuum we get (Sehweber 1964)

a* I0 > = ~/~f] a >, (6)

and ~* creates a state of I a[ particles. Similarly the action on any basis vector will be:

I / 3 +

( I/3 > --- I / 3 - >, (7)

where the last expression is defined if t3 -- a E B. Otherwise for some index i, /3z < a~ and in this case the factorial of a negative number we assume to be infinite (by identification of the factorial function with the eulerian F-function for integer numbers F(n + 1) = n! making "~ 1/3) = 0 if/3 -- a ¢ B.

Let A be the algebra generated by the {a*, ai}, i = 1,... oo by taking complex linear combinations of finite products. By (4) every finite product of operators can be written in normal form, i.e. AO's standing to the right of CO's.

There exists a one-to-one correspondence between B × B and the set of all normal form monomials, such that to a pair (a,/3) E B × B, there corresponds the

operator a*/3, and whose domain is D. Any element A E A can be written as

(~,/~)

where 2~ extends over a finite number of pairs (a,/3) and Ca/3 are complex numbers.

Together with A E A, its adjoint A* also belongs to A, then A is a *-algebra.

528 Martin Rivas

4. Representation of operators

Chamorro (1978) has shown that if A is a linear operator defined on D it admits for- mally a normal expansion written as

a = ~ co~*~, (8) (Bx~)

where the 2~ could be infinite, and the c-numbers Cap can be explicitly written (BxB)

in terms of matrix elements of A on the basis { I a ) / a E B} and they are unique. We shall prove later that (8) converges weakly to A. We merely sketch now the

calculation of the c-numbers C,,fl. In fact let us assume (8) to be a formal expansion with the C~/3 unknown. Take

a (t~, v) matrix element on both sides. The left hand side is just (g, [ A I v). Because of (7) the right hand side appears as

qa <~l~*~lv> = (Bxa)

(~xB)

and by (3) the sum extends now to those pairs (a,/3) such that/~ -- a = v -- f l e B. It contains a finite number of terms; just N~,v. Let ([~]% [/3]m), m = 1 .... , Nt~ v, be one of such pairs; if we take matrix elements ([a] m [A[ [~]m) for all m, we arrive to a linear system for the unknowns C~/~ which can be solved, giving:

(9)

where the sum is extended to the NG# pairs ([a] =, [film) which verify:

a - - [a] m~- - f l - [ff]=eB, [a]Nafl= a, ~]N,.,fl=

and the coefficients d , are real numbers obtained by means of the formulae:

dN~ = 1

tim= (-- 1)Na/~-m ([,q"! [~W.F 2

bin+l, m bin+l, m+l 0 . . . . 0

b,n+2, m bin+Z, m+l . . . . 0

bN~,. bN~, .+~... bN4j , N~-X

and the entries in the determinant are

(lO)

b,, = -~ 1 / ( [ ~ ] , - [~]')!

L 0 i f [ ~ ] , - [~], ¢ B .


4.1 Weak convergence

The normal form expansion (8) converges weakly to A. In fact, let {A.~) be a sequence of operators of A constructed as follows

n, b

A.~ = X Ca[~ ~*'~" (,,,P)

(11)

~'n, b where C~p have been calculated according to (9). and the finite sum /_,(a./~)

extends over those pairs (a. ~) such that for i>n a~ =/3~ = O, and for i ~< n ai. 13t ~<b.

Let [~> and [ ~b> be two arbitrary vectors in D. Then [~> = 27 t, ~ , I/x), [ ~b> = Z'g ~, [ v) where 27t~ and 27~ are finite sums. Put a = <~b I A [ ~b> and

n, b

(]2)

For every pair (/~, v) there exists an index Kay and an integer L~u, such that for j > K ~ v , /~j= vj = 0 a n d fo r j ~< K~,v,/~j, vj ~<Lt, ~. I f in (12)n >Kt, v , b > L ~ then this sum extends over

Kl~v, Lg

Y (,~, P)

and ifn < Kt~v, b ~<Lt~ v

n, b

it reads 4

(~,

For fixed ff and ff let K and L be respectively

K = m a x Kg~ L = m a x Lt~ v, (~, #) {~, ¢)

then the sequence {A.b) with 4, ~ arbitrary, but fixed elements of D, gives us a Cauchy sequence of complex numbers {a.b) since for n t, n~ > K, bt, bz > L, a.a b~ = a.~ b~ which is the desired limit a = (~l A 1 ~b).

We have as a corollary that the W*-algebra of bounded operators on H B, when

restricted to the domain D, is contained in the weak closure of the *-algebra A.

5. Fermionic systems

The preceding result can be extended to the fermionic case, with a few modifications. Instead of B, we define the set F of sequences of finite sum, made up of zeros and ones. F is no longer a semigroup, but can be partially ordered by writing a >f fl if

530 Martin Rivas

a --/3 E F. The number of sequences which are not greater than a given a is still O9

N~ = Iii= 1 (a~ + 1). Similarly given a pair (~,/3) the set of pairs (/~, v) which verify

that ~ -- /~ ----- /3 -- v E F is N ~ = iii_1oo {min (a,, /3,) + 1}. Let H F be the anti-

symmetric Fock space of a system of identical fermions. There is a one-to-one correspondence between F and the orthonormalised basis vectors in the occupation number representation. To every a there corresponds the basis vector [~) = [ al,... , a , . . . ) , a L being one or zero whether the corresponding one particle state fi is occupied or not.

Instead of (4), fermionic CO and AO verify anticommutation relations

[a,, aa]+ = [a*, a~]+= o,

[a,, a~']+ = ~,~, (13)

For every a ~ F, two operators are defined:

oo

~'* = II (a*) at = (a*) a~ ... (ai*) a' ..., i = l

i=1 = II (a,) a, . . . . (a,)*, ... (al)% (14)

oo

remark the reverse order in the definition of ~" with respect to (5). Acting on a basis vector [/3)

z* 1/3) =t~. l /3 + ~)

I/3) =/3~1/3 - ~) (15)

where the c-numbers fla,/3~ take only the values £-I- 1, 0} being zero when/3 q- a ¢ F. Explicitly

cO

/3,~ = II (-1)~, ~, (1 - ,~,/3,), i = l

oo

i=1

i--1 i--1

where s , = ~ /3j, s ; = ~ ( /3j--aj) . j = l j = l

Expressions (8), (9) and (10) remain valid, but now b~, is zero or one. The convergence can be similarly proved by defining the sequence of operators

n

. , .= (a, D

where ~'n (~, #) means that the sum contains only those terms such that for

i > n, ai = /3~ = 0.


6. Different kinds of particles

All the above results can be generalised to a composite system consisting of m kinds of bosons and n kinds of fermions. We can define a set B or F respectively for each kind of particle. Let G = B m × F n be the set of all ordered (m + n)-tuples of sequences. By assuming that operators defined on different kinds of particles commute, we define a Hilbert space H G as a tensor product of m symmetric Fock spaces one for

each kind of bosons, and n antisymmetric Fock spaces for the different kinds of fermions.

In the occupation number representation there exists a one-to-one correspondence between G and the orthonormalised basis vectors of H G. We can similarly define

for every a E G, a basis vector] a) E H a and a pair of operators ~* and ~', and

work out without modification that every operator A defined on the dense domain D, the linear span of the basis { I a) /a 6 G} admits a weakly convergent representation in normal form given by

A = ~ Ca/3 ~* /~, GxG

where in computing the C,t ~ only matrix elements (/~ l a i r ) which verify

a --/~ =/3 -- v 6 G are involved.

7. Vector space structure

The countable set of normal form monomials ~* ff is a basis, in the weak sense, of the vector space structure of the algebra of operators (bounded or not) defined on the dense domain D. In fact, a finite number of monomials are always linearly in- dependent. Further, if we try to represent a given one/~* ~" in a series (8), the coefficients are Ca/~ = ~at ' ~p.

From expression (9) we see that the correspondence A -7 CaB (A) is linear, since

c (ra + = Z d . ([a] ' ] rA + I

= r Zdm <[a]ml A [ [/3]m> + a Zdm <[~],[ A l [fl]m>

= rCa# (A) + sCa~ (B)

and since the arm are real in (9) we see that

Ca~ (A*) = Ca. , (A), (16)

where the bar means complex conjugation, and thus the operation of taking the

adjoint of a certain operator A is the usual when A is given by (8), A = 27 Ct~ (A) ~'*/3,

then A* = Z Ca/3 (A) /37 ~" and by interchanging /3 and a A* = 27 C/3 a (A) S*

consistent with (16).

532 Mart[n Rivas

With these properties the coefficients Ca# (,4) can be interpreted as the components

of the vector A in the basis {~*/~](~,/3) ~ G × G~-.

References

Berezin'F A 1966 The method of second quantization (New York: Academic Press) p. 21 Bogolubov N N and Shirkov D V 1959 Introduction to the theory of quantized fields (New York:

Interscienc¢) p. 103 Chamorro A 1978 Pramana 10 83 Itzykson C and Zuber J B 1980 Quantum field theory (New York: McGraw Hill) p. 111 $¢hweber S S 1964 Relativistic quantum field theory (New York: Harper and Row) pp. 185, 130,131

and 139

normal form expansion in an algebraic way

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