Equal time anticommutator structure of baryon currents in neutral pseudoscalar meson interaction

Tapper, M. L.

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THE STRUCTURE OF THE EQUAL TIME ANTICOMMUTATOR 
OF THE BARYON CURRENT 
I N  NEUTRAL PSEUDOSCALAR MESON THEORY 
M. L. Tapper 
April 1968 
Goddard Space Flight Center 
Greenbelt, Maryland 
https://ntrs.nasa.gov/search.jsp?R=19680012160 2020-03-24T00:21:06+00:00Z
CONTENTS 
Page 
Abstract . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v 
I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1 
11. Definition of the Equal Time Anticommutator of 
the Wilson Current h(x) ............................... 3 
ID. Structure of the Equal Time Anticommutator. . . . . . . . . . . . . . . . .  5 
IV. The Spectral Representation of the Vacuum Expectation 
Value(0 1 {h(x), K(x')} I O> ............................ 10 
V. A High Energy Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  16 
Acknowledgment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  17 
iii 
1 
The Structure of the Equal Time Anticommutator of the Baryon Current 
in Neutral Pseudoscalar Meson Theory 
M. L. Tapper* 
NASA-Goddard Space Flight Center 
Greenbelt, Maryland 20771 
*Research Associate - National Academy of Science-National Research Council 
2 
Abstract 
The equal time anticommutator of the baryon current h(x) which is 
the source term of the Dirac equation (ig- m) $ = igy5h(x) is studied for 
the case of a neutral pseudoscalar meson interaction. A method of calcu- 
lating the equal time anticommutator is given which incorporates the 
basic structure arising out of the Wilson hypothesis used to define the 
current. It is found that derivatives of S3 (x-x') do not appear in the anti- 
commutator. The absence of such Schwinger terms permits a theorem 
to be stated regarding the high energy behavior of the spectral functions 
p, ( M2) and p,( M2) which characterize the baryon propagator Si(p). 
_ -  
I. Introduction 
Presently much interest in the equal time current algebra proposed by Gell- 
Mann has motivated serious investigation of the properties of equal time com- 
mutation relations involving various field theoretic models of current operators. 
This interest in equal time commutation relations has led us to study the struc- 
ture of the anticommutator { h(x) , c(xf)) of the baryon current in a neutral 
ps eudos c alar meson theory. 
We make use of the hypothesis of K. Wilson2 to define the appropriate cur- 
rent operator h(x) that should appear in the renormalized field equation 
(iiJ - m) $(x) = igy,h(x) . 
We then use this operator to compute the equal time anticommutator by a power- 
ful method which has recently been applied to calculate commutators of the 
electromagnetic current in quantum electrodynamics .3  We find that the anti- 
commutator { h(x) , F(xf)) contains no derivatives of 63 (X-X~).  - -  By trying to 
understand the absence of such terms we investigate the most general proper- 
ties of the vacuum expectation value (01 {h (x), G(x')) I 0). This analysis leads 
us to the conclusion that the nonappearance of derivatives, in the equal time 
anticommutator implies the existence of a theorem which states that the spec- 
tral functions p1 (M2) and p2 (M2) which describe the propagator Sk @) must 
approach zero in the high energy limit. 
1 
In Sec. I1 we define the method for  calculating our equal time anticommutator. 
In Sec. Ill we outline the calculation and discuss the structure of the anticommu- 
tator. In Sec. IV we make a general analysis of the vacuum expectation value of 
the equal time anticommutator in terms of its spectral properties and in Sec. V 
we derive the theorem involving the high energy behavior of p1 and p, . 
2 
11. Definition of the Equal Time Anticommutator of the 
Wilson Current h(x) 
According to the hypothesis of K. Wilson2 the product 4 (x) d(x- 6) of a baryon 
and neutral pseudoscalar meson field in which 6 is a space-time variable near 
the point x, is given by the expression 
as 6 - 0. E, (e), B, (e), B, ( 5 )  a re  matrix functions in the spinor space of the 
baryon field and have singularities at the point E = 0. : $6: denotes the general- 
ized wick product and is defined solely by Eq. (2.1). If we denote h(x;5) by 
E,-'(,")[$ (x)+ @-e) - B, (CNJ(X)  - B,, ( 4 )  W$(x)] then the baryon current ap- 
pearing in the field equation4 
is given by the relation 
h(x)  = lim h(x  ; 6) . 
5-0 
Brandt has proved that Eq. (2.2) with the source current h(x) given by Eq. (2.3) 
is valid to all orders in the sense that the renormalized integral equations which 
it implies a re  derivable from and equivalent to renormalized perturbation theory? 
In addition to Eq. (2.3) we d s o  have the relation 
ht(x')  = l i m  h t ( x '  ; E ' )  I 
e'-0 
3 
We now compute the equal time anticommutator { h(x), ht (x')} according to the 
prescription 
where we evaluate the right hand side by using the following h o w n  equal time 
commutation relations for the renormalized fields # and 4 
4 
I11 Structure of the Equal Time Anticommutator 
If we denote the matrices E;'(E) B, (5) and E;, (5) B,, (5) by C; (4) and 
C 2 p  ( E )  = , C2) respectively then { h(x), ht (XI)} may be written as the limit - 
taken in  a rotationally invariant manner. Q,  Q - a re  q number matrix functions 
and S, 2, S' are c number functions given by the expressions 
- C20(5)  K1(4"') E(4"' ) E i l ( t '  ) +(x - 5"') +(x' - e' ) 

and A, B, - C ,  D, E ,  F, G, H are the following functions of E i l  , C; and C, 
B = i [my,  -t i g ~ ; +  y5yoI 
E = A t ,  F = B t ,  G = C ,  t H = D  t . - -  
The reason for this rather complicated structure is due to the presence of the 
time derivative 4 which appears in the expression for h(x) and h' (x'). We have 
computed all commutators involving $ by making use of the field equation (2.2) 
and its adjoint. 
In order to simplify Eq. (3.1) we shall relate the subtraction coefficients 
E3-', C; at 6 = 0 to the renormalization parameters Z ', Z, , Z and 6 m= m - q 
by the following method. We consider the unrenormalized field equation 
n 
and rewrite it as usual in terms of the renormalized fields $ and 4y the physical 
mass m and the renormalized coupling constant g = z;’ Z 2  Z1’2go. We find Eq. 
(3.4) to assume the form 
On comparing Eq. (3.5) with Eq. (2.2) we deduce the relations 
Zl  
2 2  
E i ’ ( 0 )  = - .  1 
(3.6a) 
Owing to the existence of the derivative term C 2 p a p $  appearing in h we must 
further introduce a new subtraction constant C2 defined by the condition 
C2, ( 0 )  1 ( C 2 ~ 5 ~ o i  - C 2 ~ 5 1 / )  (3.6b) 
where C2 is in general a complex number. We also note that Eq. (3.6b) ex- 
presses the only form C2, (0) may assume consistent with the requirement of 
rotational invariance. Using Eq. (3.6) to evaluate Eq. (3.3), the matrices A, B, 
- C ,  D, E,  F, - G, H are found to have the form 
By employing Eq. (3.7) we are formally able to evaluate the terms Q, S, Q, S, S' 
which appear on the right hand side of Eq. (3.1) in the limit . We find 6+ 0 6'40 
E"+ 0 
6"' + 0 
that the functions Q, and S' vanish, which means that the equal time anticom- 
3 mutator { h(x) , ht (x')} contains no derivatives of 6 (5 - & I )  , o r  so called Schwinger 
terms. This situation is to be contrasted with the result established for equal 
time commutators of the electromagnetic current in electrodynamics in which 
the existence of Schwinger terms has been demonstrated explicitly? 
We shall now investigate the most general conditions relating to the proper- 
ties of { h(x), h' (x')} that can lead to our result. 
9 
IV, The Spectral Representation of the Vacuum Expectation 
Value (01 {h(x), K(x))> I O> 
We have shown that the equal time anticommutator is given by the expres- 
sion 
1 - {h(x) ,  h(x ' ) )  = - bl z3(5- 5 ' )  : +,(x) : 
z2 
where M, , M, , M, are the matrices 
and J(0) is related to the meson spectral function p ( p 2 )  by the equation 
m 
J(5) = i dp2 p ( p 2 )  4. ( 5  ; p 2 )  . (4.3) 
0 
Now let us take Eq. (4.1) between the vacuum state. By the properties of the 
Wick product : +,(x) : and because +(x) is a pseudoscalar we find the vacuum 
expectationvalue to be of the form 
10 
Consider the vacuum expectation value in general, that is, before any equal time 
limit is taken. If we use Eq. (2.2) and its adjoint the general expression for the 
vacuum expectation value may be written as 
where (01 {$(XI , F(x')} I O> is just the vacuum expectation value of the anti- 
commutator of the underlying baryon field and satisfies the well known spectral 
representation 
in which A (x - X I ;  M) is the function 
d3k ( e - i k '  (x-x')- e i k '  ( x - X I )  
(2+2 uk 
A(x - x' ; M )  = - i 
with k, related to wk by k, = wk - m. P1 and P2 are scalar functions 
which have the well established properties 
p1 (M2) and p 2 ( M 2 )  are real , 
pl(M2) 2 0 , 
MPl(M2) - P2(M2> 2 0 
( i) 
(ii) 
(iii) 
11 
and the spectral integral which appears in Eq. (4.6) begins at ml , the threshold 
of the continuum spectrum. On calculating the right hand side of Eq. (4.5) we 
find that (01 {h(x), h(0)) I 0) becomes 
2 2 with the variable change M2 = a, al = ml and c = m . 
Now extreme caution must be exercised in  computing the equal time limit ! 
I We do this by regarding <O I { h(x), E(0)) i O> as an improper function o r  dis- 
tribution which we shall denote by F(x, t). We therefore introduce a sequence 
of testing functions 4 (x) f (t) such that the limit 
n '01 
f n  ( t )  = 8 ( t )  
holds and define <F(x) , 4 (x) - f (t)) by the relation 
Then the equal time limit F(x , 0) will be given by the prescription 
12 
To facilitate the calculation of F(x, 0) we introduce the Fourier transform 
4 (p) in (po) of 4 (x) - f n  (t) by the usual expression 
(4.10) 
<F(x), 4 (5) fn (t)) may then be written as 
where 6(k) is the fourier, transform of F(x). Further introducing the functions 
7 ~ :  (a) (i = 1, 1/2, 0), .ri (a) (i = 0 , l )  and E: (po), ?,O(po) by the relations 
7-r: = apl(a), = v 5  p,(a>, 77; = p,(a> , 
(4.1 la) 
and 
(4.11b) 
-0 fn(P,> -' f n ( -  Po) 
2 fn(P,> = 7 
and employing Eq. (4.7) we have the useful result 
13 
1 
(4.12) 
-J 
From Eq. (4.12) we secure the desired equal time expression for F(x, - 0) namely, 
OD 
- 1 1  --.- [ ( K t l  t cK",- 2 & K y , )  70(p)r83(5) 
r = O  
z2 g2 
ry 
where K;'j , K h j  have the form 
(4.13) 
If we choose a sequence f,(t)  symmetric in t and if we demand that Eq. (4.13) 
be consistent with Eq. (4.4) then the following condition 
K,'l + c K o  r l  - 2 & K f 2  = 0 (4.14) 
must hold for all r > 0. Eq. (4.14) implies the very important result that the 
integral 
I = [ ( a  t c) pl(a) - 2bcp2(a) 1 da  
a l  
must be finite. 
15 
V. A High Energy Theorem 
We now use the fact that I is finite to show that the spectral functions (a + c)p, 
and p, approach zero as a = M2 tends to infinity. The most general behavior of 
p1 and p2 compatible with the results of the previous section is given by functions 
of the form 
(a f c> P, = D(a> f f (a>  
2& p2 = D(a)  f g(a) 
where D(a) approaches a non-zero constant or  infinity and both f(a) and g(a) 
tend to zero as a approaches infinity. The difference h(a) = f - g approaches 
zero and is an integrable function giving the finite expression I. In this situation 
we have the following result 
2&p2 - (a + c )  p1 > o a s  a - a .  
If we recall that c = m 2  < M2 = a then the assumed behavior of P, and p2 
(Eq. (5.1)) implies the condition 
which is a contradiction! This inconsistency arises from the fact that D(a) was 
assumed to approach a non-zero constant or  infinity. Therefore D(a) must be 
zero and hence the functions (M2 + m2 ) p1 (M’) and p2 (M2) approach zero as M 2  
tends to infinity. 
16 
Acknowledgment 
The author wishes to express sincerest thanks to Dr.  Richard Brandt for 
suggesting this problem and for offering helpful advice and 'encouragement 
throughout the course of this work. 
17 
References and Footnotes 
1. Whenever we write { h(x), c(xl)> we shall always mean x, = xo'unless ex- 
plicitly stated otherwise as in Sec. IV. 
2. K. Wilson, unpublished. 
3. R. A. Brandt, University of Maryland Technical Report No. 727, August 1967 
(to be published). 
4. We adhere to the following notation of Bjorken and Dre l l  
- - g,, = - g,, = - g, = 1, y =  (7') , goo - - 
$ = y  - + y * V ,  a y , = ? = i y ~ y 1 / 3 ,  . .. 0 1 2  
O a t  - 
t t Yo = Yo, Y = - 71 {Yay Y p )  = 2g,p * - - 
5. R. A. Brandt, Annals of Physics: 44, 221-265 (1967). 
6. Bjorken and Drell ,  Relativistic Quantud Fields, McGraw-Hill Book 
Company, N.Y. (1965). 
18 


The structure of the equal time anticommutator of the baryon current in neutral pseudoscalar meson theory

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