Download 241 Quantum Field Theory in terms of Euclidean Parameters

241
Progress of Theoretical Physics, Vol. 21, No.2, February 1959
Quantum Field Theory in terms of Euclidean Parameters
Tadao NAKANO
Department of Physics, Osaka City University, Osaka
(Received October 4, 1958)
The quantum field theory is presented in terms of Euclidean parameters. It is shown
that this theory is equivalent to the conventional one in terms of the Minkowski parameters,.
if we confine ourselves to the case of local fields, and it is suggested that the former is more
adequate to describe an internal structure of an elementary particle than the latter. There
is no necessity that the parameter space of field theory is Minkowski, because it has no
direct connection with the macroscopic space. It is enough that expectation values of observable quantities are covariant under Lorentz transformations. The theory shown in this
paper satisfies this requirement.
§ 1. Introduction
In the conventional quantum field theory the parameters of the Minkowski
space are used to describe behaviors of field variables. This is based on the fact
that the macroscopic space we live in is the Minkowski one. But the parameters
in field theory are not the quantities that have direct connection with the coordinates
of the macroscopic space. The former are connected with the latter only through
field variables. Under these circumstances there is no necessity to use the Minkowski
parameters to describe microscopic states.
It is enough that the expectation values are covariant with respect to the
Lorentz transformations, and it is already shown that the internal structure of an
extended particle is described by the Euclidean parameters, but expectation values
of observable quantities are covariant with respect to Lorentz transformations. l )
In this paper we propose to use the Euclidean parameters to describe microscopic states and show that this theory is equivalent to the conventional one if we
confine ourselves to local fields.
When we consider the internal structure of elementary particles, the Minkowski
space is very inconvenient, because invariant domains are unbounded. On the
contrary, the Euclidean space seems to be adequate to describe the internal structure, as invariant domains are bounded. If the field theory in terms of Euclidean
parameters and the theory in terms of Minkowski parameters are equivalent to each
other to describe the local fields and the former is more adequate to describe the
structure of elementary particles than the latter, we should conclude that field theory
should be reconstructed by the use of Euclidean parameters.
Possibilities that we can use the Euclidean parameters instead of the Minkowski
T. Nakano
'242
ones have so far appeared in several aspects. On the one hand, tensors and spinors
are proper to the Euclidean space, but on the other hand, it is expansors2) and
expinors!" that are proper to the Minkowski space. From the fact that we know
no physical quantity which is represented by an expansor or expinor, it seems enough
to use the Euclidean parameters exclusively. Besides, the fact that the F eynman
integra14)
s~~
IS
f
co
dk ,\' dk o (kZ -ko2 + L-ic)-n
equal to the integral with respect to the Euclidean parameters
OJ
f'
i
f'
j dk.1
dk q U£2+1::l+L)-n
and the fact that the internal motion of two bound particles IS described by the
Euclidean parameters5) seem to suggest the possibility of constructing field theory
with the Euclidean parameters.
In § 2 we investigate the analytical properties of vacuum expectation values
and show that, if we know the vacuum expectation values in terms of Euclidean
parameters, we can derive the ones in t~rms of Minkowski parameters, and vice
In § 3 the transformation properties of field variables described by the
versa.
Euclidean parameters in accordance with the macroscopic Lorentz transformations
are discussed. In § 4 the variational principle is displayed and it is shown that the
transformation properties of the energy momentum and charge are same as the
conventional ones.
In § 5 and § 6 we discuss a scalar and a spinor field, respectively, and show that the quantization method and the eigenvalues of the energy,
momentum and charge agree with the ones of the customary theory in terms of
the l'viinkowski parameters.
§ 20
Properties of vaCUUlll expectation values of time-ordered
field variable products
The properties of vacuum expectation values of field variables were throughly
investigated by \Vightman. 6 )
In this section we discuss the analytic properties of
vacuum expectation values of time-ordered products of field variables. 7) For simplicity, we treat the case of a neutral scalar field interacting with itself in this section.
In succeeding sections we shall treat more general cases.
First, let us consider the product of two variables SO (X1' t 1) and SO (X2' t2)'
The vacuum expectation value is given by~)
(,[fa, SO(x 1 , t 1 )SO(X 2 , t 2 )
1jJ~)
=iLl/CCI)
= -iJ/(-) (xz-x j
,
(x j -x 2 , t 1 - t2 )
t 2 - t1 )
(2 ·1)
Quantum Field Theory zn terms of Euclidean Parameters
243
where*
00
(x, t) =
J'(±)
J
t) p (rn 2 ) d (rn 2 )
J(±} (x,
,
o
J(±)
(x, t)
= =r=-, . ~.,3
(27r)
rdk (2E)
J
/
-1
..
....... ,
exp!'il£x =r= iEtJ
",'
(2·2)
'
E=v k 2 +m'L
and l]J'o is the true vacuum state vector. The relation (2·2) shows that J(+) (x, t)
is analytic when 1m t < 0, and J(-) is analytic when 1m t> 0. Thus, if we introduce the notation
(2,3)
zo=cos{}·t-isinH·x4 ,
°
where t and X4 are real, J(t) (x, ,zo) and Je-) (x, zo) are analytic when X4> and
X4 < 0, respectively.
And d(±) (x, zo) is derived from LI(±) (x, t) by the relation
LI~±)(x,
zo)=Z(t,
O)L1(±) (x, t)
X4;
= =r= ·(-2~S;:)J dl£(2E)-1
In
exp[ikx =r= iEzoL
(2·4)
which
(2 ·5)
and
Ot=%t,
°4=O/OXq •
Especially when (}=rr/2, and xq>O forL1(i') and x 4 <0 for L1<-\L1(±)(x,zo) results
in (see Appendix 1)
J(±)
(XI},) =
j(±)
(x, -- ix4 )
-
(0
= =r= __ z.. -:;; \ dl£ (2E)
(2rr) J
= =r=
_1
expU/rx :F
~('~-)4 J" cf4 lzefPVlzl)'~I;;\,
211'
fZ!L I<\J,
Ex,,·1
°
:1:4
+ nl~
where
1
d
OJ
4
/;:
=
OJ
00
Jdl<j J' dk2 Jdk;; Jdk
-00
*
The units
n=c=l
ro
are used throughout this paper.
4
(2·6)
244
T. Nakano
and
k.,;,X(J>=kl Xl + k2 x 2+ k 3 x 3 +k4 X4'
We call the set of real ~uariables (xu X 2 , x 3 , X4) Euclidean parameters
and the space, whose coordinates are given by (xu x 2 , x 3 , x 4L Euclidean
parameter space throughout this paper.
When interactions are absent, p (x, t) is given by
'Po (x, t)
= V- 1 /2 ~k C2E)-1/2{SOoCk) exp[i(kx-Et)]
+Po*(k) exp[i( -kx+Et) J}'
(2 7)
0
where V is the normalization volume, and PoCk) and 'Po*(k) satisfy the following:
commutation relations.*
[Po (k), 'Po"" (k') J- = (~klrl
[SOoCk), 'PoCk')l-=['Po*(k), Po*(k!)J~=O,
(2 8)
0
in which
(}kkf
=
{I,
when k=k'
10,
when
l"£~k'.
When one operates 'Po (k) on the vacuurn state, it vanishes :
'Po(k)
ifJ~=O.
(2 9)
0
The summation bk extends to unlimitedly large values of k, so that p (x, ze) has
no analytic form in general. However, cP (x, zo) is an operator and it is sufficient
for our practical calculations that matrix elements of SO (x, zo) or products of p (x,
zo) 's are analytic in some regions of Zo. For example, one body amplitude
(2 10)
0
is analytic everywhere in the ze-plane except as X4---:> - co. (j)o and (j)k are the free
vacuum and one body state vectors, respectively. (See Appendix II). Thus we
may write CPo (x, zo) as
SOo (x, zo) =Z (t, x 4
;
f)
SOo (x, t)
f(
= V-l/ 2 lim ~k (2E)
-1/2
{q;o (k) explJ(kx- EZe) l
J(-"?CXl
+ Po* Ck) exp [i (kx + Ezo)j) ,
(2 ·11)
K
here 2JT~ indicates the summation up to \1£\ =1<:. and the limit K---:>co must be taken
after completion of calculations. By the use of the relations (2 8) and (2·9) with
expression (2 ·11) we get
0
*'
The bracket symbol is defined by [A, B] ±=AB±BA.
Quantum Field Theory in terms of Euclidean Parameters
245
Ie
= V-I lim 2.~k (2E) -1 exp[ik(x1-x Z) -iE(zw- ZZ9)]
K-'?oo
= (2 7r
)-:'J
dk(2E)-1 exp[ik(x1-x Z) -iE(Z10- ZZ0)]
=iLl(+) (x1-X Z ,
(2 ·12)
ZIS-Z20).
When (}=7r/2, (2·11) and (2·12) become
q;(x IL ) =q;(x, -iX4)
I(
= V- I / 2 lim ~1" (2E) -1/2{q;(k) exp[ikx--Ex4]
K-'?oo
(2 ·13)
and
(2 ·14)
Next we consider the vacuum expectation value of the time-ordered product of
two field variables
(2 ·15)
where
LlJ (x, t) =
,
I
2iLl'(+) (x, t), when
t> 0
-·2iLl'(-)(X, t), when t<O.
(2 ·16)
Then
LlJ (x, zo) =Z (t, X4 ; (}) LlJ (x, t)
is analytic in the whole zo-plane except the
line X4=0 and Ll~r/ (x, t) is the boundary
value of the analytic function LlJ (x, ze) approaching the line X 4 = 0 in such a way that
Zo gets near to the positive t axis from the
half plane where X 4 is positive and to the
negative t axis from the one where X 4 is
negative (see Fig. 1).
vVhen (} = 17/2, we get
-------.~--~~~L-
____- - - - t
Fig. 1. The directions of continuation
to get 11]/ (x, t) from .h/ (x, ze).
Llr' (x IL ) = LlJ (x, - iX4)
=
(2:)') d(m') Jd4k"~;-1~~;:21p(m2)
(2 ·17)
246
T. Nakano
The relation (2 17) is also obtained as the vacuum expectation value of field
variable product ordered according to the values of X4' s :
0
(lffo, P (SO (X 1JL )
=
SO (X ZJL )
,
)
([fo)
Wr),
1(IP;, SO (X:w) SO (X]JL) ?jJ~),
r (lffo, SO (x 1JL ) SO (X ZfL )
iLi!C+)
1
-
(X 1JL -
. A!(-) (
ZI.J
XZI'),
X1!J_ -
X 2P,
)
,
> X 24
when
X 14
when
X Z4 > X]4
when
X]4 >XZ4
when
X Z4 > X]4
=~ Lip! (X]fL-X2rJ.
In general we consider the vacuum expectation value of the time-ordered product
of n field variables:
(2·19)
Then, if we fix the order of values of
comes the Wightman function6)
Fen) (x]' t 1 ;···
;
X n,
t], . ", tn
t n) = (ljJ~),
SO
as, say,
t]
> ... > tn, (2 19) be-
(x]) t])··· SO (xn' t n) Wo) .
0
(2·20)
From the analysis of Wightman we know that*
(2·21)
IS
analytic when
F en) (Xl'
- Z X 14 ;
In this way if we know the values of
(lFo, P(cp(xuJ, "', cp(xnrJ)lfFo)
we can determine the values of (2 ·17), and vice versa.
~
3.
Lorentz [ransformations
In this section we discuss transformation properties of field variables with the
Euclidean parameters.
\Ve know that the spinor representation of the Lorentz transformation is obtained by the use of the unitary trick** from a unitary representation of rotation
* Here we omit
** (p-f = </J*r.1 .
the suffixes fh,'s of
Zk.
Quantum F£eld Theory £n terms of Eucl£dean Parameters
247
In the Euclidean space. As our parameter space is Euclidean, it is necessary to
use the same technique as that we use to obtain the spinor representation.
Consider an infinitesimal inhomogeneous Lorentz transformation in macroscopic
Minkowski space:
(3 ·1)
where
(lJ1J:V
and c!.t are infinitesimal parameters and the (I)wv's satisfy the relations
vVhile fk, (lJ1cl and c1c, (k=l, 2, 3) are real, f4=ifo, (I)1c4=i(lltco and c4 =£CO are pure
imaginary numbers. As XIJ,' s are parameters, they should remain unchanged under
the transformation (3 ·1). Field variables, however, are transformed in the followIng way:
q'(Xl!)
=).((0,
c)A((t)q(x!.t),
(3·2)
In which q(xl!) is a field variable and
(3·3)
and
A((/) =1
+ i/2
W(J-"
(3·4)
Sf""
spin angular momentum matrix.
On the other hand, the Hermitian conjugate of q(x) is transformed as
q'* (XI.,) =).* (0), c) q* (XI.,) A* (0),
where
(3 ·5)
(3·6)
and * means Hermitian conjugation for matrices and complex conjugation for ['numbers. As (3·5) and (3·6) are not inverse operators of (3·3) and (3·4)
respectively, we need to introduce an adjoint field of q(x) (see Appendix III)
(3 ·7)
where
X
and D satisfies the relations
I)' r -
(x 1 ,
X
2,
X
3,
-
'Y' )
""-"4
248
T. Nakano
(k,I=1,2,3),
Then q-r (Xp.) is transformed as follows:
qlf (Xp.) =A (w, c) qi (xII-) . A-1 (w),
(3· 8)
in which
A-l(w) =1-i/2wp.lIS~1/'
By the use of the relations (3·2) and (3·7) the following formulae are obtained:
Jd xq't (Xp.) q' (x,,,)
= Jd xA (lV, c) (X
(w, c) q(X
= Jd xqf (x,J
c) A(w, c) q(x[J.)'
Jd xqt (x[J.) q(XfL)
4
l' =
4
qf
4
IL
) .).
IL ).
).-1 (0),
(by partial integration)
4
=
=1,
Dp.' =
=
=
(3·9)
Jd xq (XiLroiLq' (x!B.)
Jd xqt(XiL»).-1(W, C)OIL).(W, c)q(x,,,)
Jd xqt (x iL ) °lLq(X.,J +w\h" Jd xq(XiL ) o"q(xiL )
4
lf
4
4
4
(3,10)
Jd 4xq'f (x lL ) xiLq' (x lL )
= Jd xqt (X iL ) (w, c) XfhA (w, c) q(x
X/ =
4
;'-1
lL )
= Jd 4xqt (X iL ) xfhq(x,J + (ViLli Jd 4xl(x fL )X1)q(XiL ) +c IL Jd 4X qt(XfL)q(XfB)
=XIL + (Up." Xli + ciL I,
where
:and
(3,11)
Quantum Field Theory in terms of Euclidean Paratneters
It is easily shown that 1, D4 and LYle are Hermitian operators and
anti-Hermitian operators: for example,
I)le
249
and X 4 are
('
{J d x qi' (xrL ) q (X\L) } '*
4
1'* =:
Jd
= Jd
=
4
x q'* (xrJ q (x/)
4
xq'* (xrLI) q(X\L)
=1,
('
• '* ()
X 4'* -- J\ d 4:rq
X[L x 4 q (xtrL)
=
j' d x qt (X\L) X/ q (XrL )
4
=-X4 •
For space reflection
(3 ·12)
q ( X)
IS
transformed as
(3 ·13)
where P
with D:
IS
a unitary space reflection matrix for spm variables and commutable
PD=DP.
Thus
('
= J d x(/(x[L)q(xrJ
4
=1,
D Ie ' =
':\ ( -X[L'")
Jrd 4xq+( -X\Lt) ulcq
etc.
For time inversion (mathematical)
r;le' = r; k ,
r;l
=-
r;-4
(3·14)
250
q (x)
T. Nalwno
IS
transformed as
(3·15)
where T is a unitary time inversion matrix for spin variables and commutable with
D for integral spin fields and anticommutable with D for half odd integral spin
fields. And for transformation (3 ·14) the order of field variables in products is
assumed to be reversed. Thus, for example,
I , = ==I
=
=-1::
Jr{I4 x q (X IL i-);q (X
r_t
r)
j' {f'lxq(xrJ qi (x,J
=1
(3 ·16)
where + is taken for integral spin fields and - for half odd integral spin fields.
Relation (3 -16) implies that integral spin fields should be quantized according to
Bose-Einstein statistics and half odd integral spin fields should be quantized accordmg to Fermi-Dirac statistics as they are in the ordinary Minkowski variable theory.
The charge conjugate field of q(x) is given by
(3 ·17)
where C is a charge conjugation matrix for spin variables. Thus, a neutral field
variable is not a real function but a self-adjoint function; for example, a neutral
scalar field SO (x) satisi-ies the following relation:
~CJi
~
4.
(x) =
so* (xi) =q; (x) _
(3 ·18)
'r'be vmeiational principle
We consider a Lagrange function L which depends on any c-number functions
q(x) of the Euclidean parameters x lJ . and their first derivatives
(4 ·1)
as well as their adjoint functions {/ (x) and g'L (x) =OILq' (x) but which does not
depend explicitly on xv,. The variation principle
i
-
(~ j' L (q,
giL; qf, gl},i) d 4 x=O
(4·2)
which the variation has been assumed to vanish on the boundary of the domain
of integration determines the field equations
111
and
(4·4)
Quantum Field Theory in terms of Euclidean Parameters
251'
An energy momentum tensor can be formed from the Lagrange function as
(4·5)
which, because of (4·3) and (4·4) satisfies the continuity equation
(4·6)
If we assume that the Lagrange function L is invariant with respect to a phase"
transformation
and
where a is a constant number, differentiation of L with respect to the phase agIVes the relation
(4·7)
Then a current vector defined by
(4 ·8)
satisfies, on account of (4·3), (4·4) and (4·7), the continuity equation
CJfLjfL =0.
(4·9)
The three-dimensional volume integrals
P fL - i IJTfL4 dv
-
(4 ·10)
and
f'
Q= -ie J j4 dv ,
(4·11)
dv = dXl dX 2 dX3
give the energy momentum four vector and charge respectively and they are independent of X 4 owing to (4·6) and (4·9) respectively:
CJ 4 P iJ_=CJ 4 0=0.
As the scalar and vector properties of (3·9), (3 ·10) and (3 ·11) with respect
to the transformation (3 ·1) are derived with use of partial integrations of the fourdimensional integral, the vector and scalar properties of P!1- and 0 which are given
by three-dimensional integral cannot be shown in the same way as before.
However, owing to the continuity equations, we can show their transformation propertiesas follows.
With respect to the transformation (3 ·1) q~ and qfLi are transformed as
:252
T. Nakano
q"./ = 0 1Li. ( (0, c) A (w) q
=1.((1}; C);'-l (I), c) 0/hI. (w, c)A(w)q
= I. (w, c) (aiL
+ w :.,I:1,,) A (I)
p
(4 ·12)
q,
qfL,t=OILi.((I), c)qt·A-1(I)
=1.((1), c) (0(.l-+W!L"O,,) qi-. A-I (r).
(4·13)
Then oL/oqfL and oL/oq/ are transformed as
oL' /oq"./ =1. (I), c) CoLloqlL + (o!L"oLloq,,] . A-I (w),
(4 ·14)
oL' /oq/t =A ((I), c);l (I» [aL/oqjhi- + WILY oL/oq/J .
(4·15)
Thanks to (3·2), (3·8), (4·14) and (4·15), we get
/4=i[aLI/ oq4' ql_q/i- oL' /oqP]
=
A(m, c) {j4 + O)4kj k} •
(4·16)
Putting (4 ·16) into (4 '11), we have
QI=-iej'j/dv
('
j
= -ie dv[j4+w4kj1c+ (O)1c4Xlc04-C404)j4]
('
=-iej dv[j4+(I)q'cj1c-(W1c4Xk-C4)Oljl]
Jdv [j4 +W41cj1c + W!c4j!c]
-ie Jdv},.
= -ie
=
(owing to (4·9»
(by partial integration)
=0.
Thus we know that Q is a scalar quantity.
four vector in the Minkowski space.
§ 5.
Similarly, we can prove that P rL is a
A scalar field
In the ordinary Minkowski parameter theory SO* (x, t) SO (x, t), for example, is
-a positive definite quantity, but in our Euclidean parameter theory the quantity
corresponding to it is given by SO+ (xIJ SO (XfL) which has in general no definite sign.
In these circumstances it is interesting to investigate the signs of the energy mo_mentum four vector and charge.
The simplest example is the scalar equation without any interaction
Quantum Field Theory in terms of Euclidean Parameters
in which 0
253
is the operator
D=0iJ,0fJ-=O/+O/+O/+O/.
(5·2)
The wave equation (5 -I) can be deriv'ed by the variational principle (4·2) if
we use the Lagrange function
(5·3)
where <pf (x) =SO* (xl").
From this we get for the energy momentum tensor by eq. (4·5)
Tv,v:::= OIL SOl - 0", SO + d", so 0fJ- SO - rJ IL ", L
1- •
'(5·4)
and for the current vector by eq. (4· 8)
j fJ- = i [d 11, sot . <p - <pt
° SOJ.
(5 ·5)
fl.
The solutions of (5 -I) and its adjoint equation are written as follows:
T(
SO (xlJ
= V-1[2lim ::8", (2E) -1{2 {SO+ (k) exp[ikx- Ex
4]
so- * (k) exp[ -ikx + Ex J} ,
4
I{--)-oo
(5·6)
I(
SO' (XiJ,) = V- 1[2lim ::8/r- (2E) -1[2 {SO+* (k) exp[ -ilfX + Ex4J+so- (k) exp[ilfx- Ex4 ]L
I(-,;oo
(5 ·7)
where
Putting (5 -6) and (5 -7) into (5·4) and (5·5) and using the definitions (4·
10) and (4· 11), we have for the total energy, total momentum and total charge
Po= -iP4 = -
J
T44
dv= ::8/.,E[<p+ * (k) so+ (k)
+<f- (k) SO-* (k) J,
(5· 8)
+ SO _ (If) SO _* (k) J,
(5 . 9)
Q = e :Sir l <p + * (k) SO + (k) - SO _ (k) SO _* (k) J.
(5 . 10)
p = 2..~k k [<p + * (k) SO + (k)
These expressions coincide with those of the ordinary theory described in terms of
the Minkowski parameters. Then we can quantize the scalar field according to
Bose-Einstein's statistics, that is, we may put the commutation relations
(5·11)
;and the other commutators vanish.
From the relations (5 ·11) we find that the eigenvalues of
.IV+ (k) =<P+ * (If) SO+ (1£),
N_(1£) =SO-*(1£)SO-Ck)
(5 ·12)
254
T. lVakano
are the positive integers or zero.
When we replace the c-number relation
by the q-number relation
1/2 (cp+ Fcp + cpFcp+) ,
(5·8) to (5·10) can be rewritten as
Po =
:2"':", E [N + (k) +N _ (k) + 1] ,
p= )::d"k[N+ (k)
(5·13)
+ iV_ (k) + 11,
(5·14),
Q = e :Sir [N+ (k) - N _ (k) ] .
(5 -15)
With these definitions, (5 ·11) and (5 ·13) to (5,15), we have the relations
i[cp(x[.t), PIJJ=o[.tCP(x[.t),
(5 ·16)
[SO (xl-') , OJ = ecp (xIJ.) ,
(5 ·17)
etc.
§ 6.
A spinoI' field
The Dirac wave equation in terms of the Euclidean parameters
IS
written as
(6 ·1)
by usmg the ordinary Dirac matrices, the Hermitian conjugate equation becomes as
and the adjoint equation
IS
given by
oIJ. sui- (x) r [.t -
J1l<j;i"
(x) =0,
(6·2)
where
The wave equations (6 ·1) and (6·2) can be derived from a Lagrange function
(6·4)
which vanishes if the wave equations are satisfied. Using eqs. (4·5) and (4·8)
we get for the energy momentum tensor and the current vector
((')'t" a {u _ : l (Ul r ,(')-]
T p_1J -ll- 2 _7 I \J IL 7
U 1L 7
\J 'i~ _
(6·5)
and
(6· 6)
respectively.
The solutions of eqs. (6 ·1) and (6·2) can be written as
I(
s(,(x) = V- 1/ 2 lim 2~:,.· ~ {0~·(.k)a_;(k) exp[ikx-Ex4J
J(--~w
1'=1,2
Quantum Field Theory in terms of "Euclidean Pararneters
255
-and
ej/(x)
= V-
K
1 2
/
1im ~k:b {¢!r(k)a_~'(k) exp[ -ikx+Ex4 ]
K~oo
r=1,2
+¢::.' (k)a:'(lf) exp[ikx- Ex
(6·8)
4 ]} ,
in which
f a~' (k) =a!" (k) r4'
(6·9)
la:. (k) =a"!.r (k) r4
:and they satisfy the following relations:
\ a!" (k) a; (k) =
(a; (k) a; (k) =6rs
(~"s,
1a:'(k)a~(k) =
1a"!." (k) a!. (k) = ars
(6·10)
-Ors'
and
);?,a;' (")a;'(") = (2m)
{\ ~ a:' (k) a:' (k) = (2m)
-I
(~ilq: Er4 +nz),
-1
(zkr-Er4+m).
(6 ·11)
"=1,2
From eqs. (4·10), (4 ·11), (6 -7), (6·8) and (6 -10) we get for the energy
momentum four vector and for the charge
Po= ~k:b E[¢t"(k)¢;(k) -¢:'(k) ¢-=r(k)],
(6 ·12)
1'=1,2
p= ~k:b li[¢tr(k)cf;(k) -cf::"(k)¢"!.r(k)],
(6 ·13)
r=1,2
Q=e2}., ~ [y'J~r(k)¢:'(k) +cf::,'(la:)¢"!.r(k)].
(6 ·14)
r=1,2
These relations suggest that, as in the Minkowski parameter theory, we should
quantize the spinor field according to the Pauli exclusion-principle. Thus we put
the commutation relations
" (k) ' 7(b*S
'b*S
[-dJ
(6·15)
7 +
+ (k') J
_+ = [,&
'i' -r (k) ,'1/
- (h') -]. + = (J,'s (J "'-kl
and all the remaining brackets with the plus sign of these quantities vanish.
to the commutation relations the eigenvalues of
_N:'(k) =Sb!r(k)¢;(li) ,
Owing
(6·16)
N:. (h) =y'I"!." (k) ¢:. (k)
:are 0 and 1.
Antisymmetrizing with respect to
lead to
~b
and
su*,
the energy, momentum and charge
Po= :E,~ ~ E[N;(k) +N:'(la:) -1],
(6·17)
r=1,2
p= ~k ~ k[N;(k) +N:'(k) -IJ,
(6·18)
,.=1,2
0= e ~k ~ [N_; (k) - N:. (If) J.
'r=1,2
(6·19)
256
T. lValwno
§ 70
Concluding remarks
\Ve have seen in the preceding sections that the quantum field theory in terms'
of the Euclidean parameters gives 1) the vacuum expectation values of x 4-ordered
field variable products (P-products) which can be analytically continued to the ones
of time-ordered products with the Minkowski parameters, and 2) the same eigen
values of the energy momentum and charge as those of the conventional theory.
In these circumstances we may conclude that the field theory in terms of the
Euclidecm parameters is equivalent to the customary one and gives the same values
to the observable quantities as those of the latter, if we confine ourselves to the
local theory.
\Vhen we extend our theory to include non local fields or interactions, we meet
with various difficulties arising from the incompact property of the IVIinkowski space.
But in our Euclidean theory such difficulties do not seem to occur. For example,
exp 0 is a converging factor in the Euclidean theory but not in the lVIinkowski
one. Thus, if nature requires the nonlocal theory, we may say that the Euclidean
theory is more adequate to describe nature than the Minkowski oneo In this case
the two theories cannot be continued analytically to each other owing to the nonlocal interaction.
In this paper we have been concerned mainly with the case where the parameters are Euclidean. Certainly it is most convenient to describe the internal
structure of an elementary particle.
If we confine ourselves to the local theory,
however, it is possible to construct the equivalent theory in terms of the more
general parameter zo=cosf}·t-i sinO·x4' where O<f}<7[, instead of t as one can see
from the relations, e. g., (2·21), (A· 4), (A· 5), etc.
A method for practical calculations, the connection between our theory and the
classical equation of motion of a particle, and so on, will be discussed in the succeeding papers.
In conclusion the author would like to express his sincere thanks to Prof. Y.
Yamaguchi for his encouragement and to Dr. T. Murota for valuable discussions
and reading of the manuscripL
Appendix I
ico
E'valuation of a definite integral
Jf(k
4)
exp l£k4 x 4 l (k 2 + k 4 2 + m 2 )
_1
dl:: 4
-- OJ
vVhen f(k 4 ) has no pole in the upper half
plane of k", the only pole of the int.egrand is
-ioo
Fig. 2.
=iE.
The path of the
integration.
QuantU1n Field Theory in terms of Euclidean Parameters
Thus the path of the integration is replaced by a closed loop as shown
and we obtain the formula, when X4> 0,
257
In
Fig. 2
00
J
f(k 4) exp[ik4x 4] (E 2+ k/)
-1
dk,
JfU~4) exp[£k4x 4J(E2+kl)_1 dk 4
=
(A-I)
=7r/Ef(iE) exp[-Ex"l.
Appendix II
A note for integration with resj)ect to
X4
As an example of integrations we often meet in practical calculations, let us
consider the Salpeter-Bethe equationf)) for a bound system
Sb(I, 2) = -i
1111 d x
4
3
crx4 d"x;,;c[4xG "Sp(1, 3)S],(2, 4)G(3, 4; 5, 6)¢(5, 6),
.
(A·2)
where xI./s are Euclidean parameters and G(3, 4; 5, 6) IS a vacuum expectation
value of x,,-ordered field variable products. As seen in ~ 2, the vacuum expectation
values have its Fourier transform (see, e. g., (2·17) and (2 ·18». As to ¢ (1, 2)
=(~b P(¢(l), S!;(2»lfl2- il f)d.V) , with respect to the relative coordinate X{L=XlIL-.X2IL
it may be analytic but with respect to the absolute coordinate x IL =ax11L (1·- a) X2!L
the situation is same as (2 ·10) :
1d l:¢(k) expiJk!LX[J.] explK·X-EX -1.
4
Sb(l, 2) =
4
(A·3)
Then the integral of the right-hand side of (A· 2) gives rise to the integral
of the form
00
(f)
Jd ~\/ JdK,d'( K
1=
-co
4)
exp [£1(4 (X4 - Xl) - EX/],
(A·4)
-00
III which E is the total energy.
The integration with respect to Xl, if we integrate formally, gives rise to divergence.
In this case one should PJstp::me taking
the lower limit of X/ for the integral:
00
co
('
('
1= ~~~ .~ dX/.~ dK4 f(K,,) expiJK4 (){4-)Cn -EX/]
-A
-co
00
= -lim
A-)o-oo
Jr dI(4f(K
4)
(E +i!(4) -1 exp[£K" (X4
+ A.) + EA J.
-00
When f(K4) has no displaced pole in the upper half plane of K", we have
258
T. Nakano
I= -2n:i lim {f(iE) exp[ -EjX4 +Aj +EAJ+::S gl(Wl) exp[ -WljX4 +Aj +EAJL
A~oo
l
where w/s are the values of -iK4 at the poles of f(K4) and satisfy the inequality
(vl>E. gl(W)'S are the residues of f(K4) (E+iK4)-1 at the poles K4=iwl.
As
A tends to infinity, X 4 + A becomes positively definite and we finally get
(A 5)
I= -2n:if(iE) exp[ -EX4J.
o
Comparing (A· 3), we see that we can factorize exp [ - EX4J in the both sides of
(A·2).
Appendix HI
Interpretation of the formalism in §3 according to the operator Z(t, X4; fJ)
With respect to the Lorentz transformation (3 ·1), q(x, t) is transformed into
q'(x, t) =l.t((I), c)A(w)q(x, t),
(A·6)
where
It(w, c) =I+O)klXkOl+W1cO(XkOt+tOk) -ck0Jc-codt.
Then, operating Z (t,
X
4 ;
(A·
B) on the both sides of (A· 6), we obtain
(A·8)
and
Z (t,
X
4 ;
B) At «(0, c) Z-l (t,
X
4 ;
B)
+ (Okl XkOl +WkO {Xk (cos Bat +i sin B( + ze Ok}
-CkOk-CO (cos Bat +i sin B(4) .
= 1
4)
(A·9)
Especially when B=n:/2, (A·9) is written as
Z(t,X4; n:/2»)t((/), c)Z-1(t, X 4
=)«(/),
;
rr:/2)
c).
(A·IO)
Thus we get the relation (3·2) :
q' (XfL) =q' (x, - ix4 ) =1. (w, c) A (w) q(XfL).
In the next place, operating Z (t, X 4
Z(t, X 4
;
;
rr: /2) on q* (x, t) D, we have
rr:/2)q*(x, t)D=q*(x, -ix4 )D=qT(XfL).
CA· 11)
On the other hand,
q*(x[J)D={q(x, -ix4 )}*D=q*(x, ix4 )D.
Comparing (A ·11) and (A ·12), we obtain the relation (3·7) :
qi (XfL) =q* (x/)D.
(A·12)
Quantum Field Theory in terms of Euclidean Parameters
259
References
1)
2)
3)
4)
5)
6)
7)
8)
9)
T. Nakano, Prog. Theor. Phys. 15 (1956), 333.
P. A. M. Dirac, Proc. Roy. Soc. A. 183 (1945), 284.
Harish-Chandra, Proc. Roy. Soc. A. 189 (1947), 372.
R. P. Feynman, Phys. Rev. 76 (1949), 769.
G. C. Wick, Phys. Rev. 96 (1954), 1124.
A. S. Wightman, Phys. R.ev. 101 (1956), 860.
The author was informed by private communications from Profs. Y. Yamaguchi and G.
Takeda that J. Schwinger had reported on a similar work at the High Energy Conference
at Geneva, 1958. The author is indebted to them for their kind information.
H. Lehmann, Nuovo Cimento 11 (1954), 342.
E. E.. Salpeter and H. A. Bethe, Phys. Rev. 84 (1951), 1232; K. Nishijima, Prog. Theor.
Phys. 12 (1954), 279.