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Scattering theory for lattice øDitD14 theory
Volume 122B, number 3,4
PHYSICS LETTERS
10 March 1983
SCATTERING THEORY FOR LATTICE ~)4+1 THEORY Wlodzlmlerz GARCZYlqSKI
Instttute of Theorencal Physws, Umverstty of Wroclaw, ul Cybulskwgo 36, .50.250 Wroctaw, Poland Received 17 November 1982
Feynman rules are derived for a lattice version of the q~;+l theory. The lattice values are transcribed, vm a quaslcontinual representation, into a continuous, non-local in spatial varmbles field theory, whmh Is then quantlzed by the path integral method.
A purpose of this note is to lbrmulate a perturbaUve scattering theory for a faithful lattice version of the 4 + 1 theory. For this we use, as a main technical tool, so called quaslcontlnual representation of latlme values of the field [1~2]. We refer to these papers for all the techmcal and notational quesUons. Let us start with the classmal lagranglan of the theory L = ½OuqSO/a~b ½m2~b2 - @/4!) ~b4 .
(1)
Introducing the eqmdlstant lattices on each axis, with lattice constants a0, a 1 ..... aD we pass to the lattice values of the field
(~(na), na = (n0a 0, n l a l ..... nDaD), n u @ Z} .
(2)
Performing the quaslcontmual interpolation between these values one replaces them with a single field Sa of contmuous vanablesxU (cf. refs. [1,2] and also refs [ 3 - 7 ] ) D
Oa(X) = ~I a u ~n CMn) 8a(X - na) ~ OC(a, 0 ) .
(3)
~=0
An action functional of this field is given by the formulae D
=f dxi½Ouq,a(X) OUg~a(x) _ ~m2q)~(x) l
~Xf
dx 1 dx2 dx3 dx4Ma(Xl,X2,X3,x4J~a(Xl)(Oa(X2JOa(X3J~a(X4)
(4) where 0,~b(n) are the quasicontmual derivatives, and M a are known form factors. From this we infer that the canonical lagrangmn is
_a
I
")2
LalX ) - ~Ojjd~a(X) O#Oa(X) - ~tn-~a(X ) -
X ~.. f dx 1 dx2
dx3 Ma(X,Xl,XZ,X 3) ~)a(Xl )d~a(X2) ~a(X3) ~)a(x)
= LOa(x) + Lint(x) ,
0 0 3 1 0 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
(5)
275
Volume 122B, number 3,4
PHYSICS LETTERS
10 March 1983
while the effectwe lagranglan IS of the form
f~a(X) ~OtadPa(X)~t~(Pa(X)
~m24~2(x)
dx 2 dx3Ma(X,Xl,X2,x3) (Pa(X1)~a(X2)~)a(X3)~)a(X) .
(6)
The appearance of two lagranglans xs a typical feature of a non-local field theory [8,9]. The last lagranglan determines the equation of motion of the field q~a through the Euler-Lagrange equation
-- m2) (Pa(x) = ~X. Jf d x 1 dx2 dX3Ma(X,Xl,X2,X3)dPa(Xl)dPa(X2)~a(X3)
(7)
The canomcal lagranglan L a serves for constructing a hamdtoman of the field q~a, and thus plays a central role m the quantlzatlon procedure. The quantizatxon is possible ff a theory is local m time. Thus removing a lattice from the rime axis by taking the limit a 0 -+ 0, and setting a 1 = ... = aD = a one gets for the field
Cka(t,x) = a D ~ (p(t, n) 5a(x - na) E QC(a, 0 ) .
(8)
I1
The field is continuous m the txme variable while m the spanal variables the dependence ,s of quas~contmuous character. The form factor M a becomes [1,2]
Ma(X, Xl,X2,x3)
5(xO-xO) 5(xO-xO2)~(xO -xO)Ma(XX ,X ,l2x,3)3
"
,
(9)
aO--~O
and integrations over the time variables are removed m the formulae (5), (6) and (7). According to the general theory of constructing of constants of motion for non-local theories one gets for the hamdtonian the formula [8,9] D 2 k=l [3kq~(x)]2 +~m24~2(x) +
f dDx I dDx2 dDx 3 Ma(X, Xl, x2, x3)~b(Xl)~(x2)q~(x3)~(x)),
(1 0)
where
~o(x)=-~)a(t,x)lt= 0 ,
n(x)---(~.(t,x)lr=o.
(11)
The hamlltoman xs chosen in such a way as to secure the eqmvalence of the canonical Hamdton equatmns with eq. (7). The theory admits the path-integral quantlzation procedure [ 1 0 - 1 2 ] , and yields, after necessary modifications, the following formula for the S-matrix describing a scattenng on the interaction :LaTM. + J " q~, where J(x) is an external classical source
S(J, X) = : e x p { - ¢ 0 [a+, a] K8 ~SJ} Z [J, X] : ,
(12)
and Z [J, X] = N - 1 (X) exp (l .Lmt [8/81J] '}[ao=0 exp (~ 1JAiJ}.
(13)
Here the functmn A(x - - y ) is gwen by the formula oo
A(x - y ) = (2~r)-W+l)
~f dpof -~
dDp exp[_lP0(X0 _ y 0 ) + lp(x - y ) ] [(p0) 2 - (p)2 _ m 2 + le]-1 .
~ *
s4* is a compactlficatlon of the set M of momenta
276
(14)
Volume 122B, number 3,4
PHYSICS LETTERS
10 March 1983
05)
s~ = {p;-rr/a<~Ptc<~rr/a,k = 1..... D} , A =rr/a . The free field qS0[a+, a] (x) as given by Oo[a +, a] (x) = (2rr) -D/2 f dDp {a(p) exp [-l(x0COp _o~* (2C°P)1/2
px)] + h.c.}
(16)
and the symbol Llnt . [ ] stands for the expression :Lmt[qs] =
dx dx dx dx
U
......
Ma(x,x ,x ,x '
"
)lao= 0
"'
X [ ( a ( x ) ( a ( x ' ) O ( x " ) ~ ( x " ' ) - 6 A ( x - x')c)(x")c~(x " ' )
+ 3 A(x
--x') A(x" -- x ' " ) ]
(17)
.
Taking into account the formulae co
q~(x) = (err) -(D+I)/2 f
dP0 f
(:8)
dDp ep(p) e x p ( - i p x )
(slmIlat for J(x)), and
Ma(X,X',X",x'")=(2rr)-3(D+l) f dp dp' dp" dp'"6(p +p' + p " + p " ' ) rood 2A s~Sx.~* X exp[l(px + p'x' + p . .x. . + p ' " x ' " ) ]
(19)
whe re
dl,=dPodPl...dPD ,
(20)
.~* = s ~ X ... X M~) ,
we get for the generating functional of the full Green functions
ZlJ, X l = N - ' ( X ) e x p [ - ~ ( 2 r r ) - ( ° + l ) •
f
( X
dpdp'dp"dp"'6(p+p'+p"+
+ p,) aIJ(p)61J(p'i&J(p")&J(p") (~_J
X exp
p"') rood 2A, a o =
_~ ~ x 9i,,
0
8(p + p')
6p 2 - m 2 +le 61J(p")&J(p") + 3 p 2 - m 2 +lep "2 - m 2 + le ,l-J
lS(q+q') ,j(q,)). dq dq' lJ(q) q2 _ m 2 + le
(e:)
The Feynman roles for constructing diagrams are: I
internal line.
vertex"
p
\>
-
~p,,, ~,p"
p2 _ m 2 + le '
p +p' +p" +
dp loop integration ,~rrD+-- ,
p" = 0 ~ -IX, rood 2A, a0 =0
symmetry factor S .
(22)
\
The Feynman rules in the coordinate space are more comphcated since they involve "generalized vemces" which appear under the form factor Ma(x , x', x", x " ) . 277
Volume 122B, number 3,4
PHYSICS LETTERS
10 March 1983
References [11 [2] [3] [41 [5] [6} [7] 18]
[91 [10] [11] [ 12]
278
W Garczyfiskl and J Stelmach, J Math Phys 22 (1981) 1106. W Garczyfiskl and J. Stelmach, J Math. Phys 23 (1982) 1168 K. Wilson, Phys. Rev. D10 (1974) 2445 S.D Drell, M Wemsteln and S Yanklelowlcz, Phys. Rev D13 (1976) 3342. D. Rogula, Bull Acad. Polon. ScL, s~rle Scl. Techn 13 (1965) 337; see also- Quaslcontmual theory of crystals, Arch Mech 28 (1976) 563. J.A. Krumhansl, Generahzed continuum field representation for lattice vibrations, in Lattice dynamms, ed R.F. Walhs (Pergamon, New York, 1965) p. 627. I A. Kunm, Pnkl. Mat. Mech. 30 (1966) 542; see also Teona upruglkh sred s mlkrostrukturol (Nauka, Moscow, 1975) [m Russian] J. Rzewuskl, Field theory, Part I (Polish Scientific Publishers, Warsaw, 1964) Ch I1. W Garczyfiskl and J. Stelmach, Bull. Acad Polon., C1. IIl 28 (1980) 141. J Rzewuskl, Field theory, Part II (Pohsh Scientific Pubhshers, Warsaw, 1975) L.D Faddeev, Les ttouches Lectures, Session XX (North-Holland, Amsterdam, 1976). D J Amlt, Field theory, the renormaltzatlon group and critical phenomena (McGraw-Hill, New York, 1978).
PHYSICS LETTERS
10 March 1983
SCATTERING THEORY FOR LATTICE ~)4+1 THEORY Wlodzlmlerz GARCZYlqSKI
Instttute of Theorencal Physws, Umverstty of Wroclaw, ul Cybulskwgo 36, .50.250 Wroctaw, Poland Received 17 November 1982
Feynman rules are derived for a lattice version of the q~;+l theory. The lattice values are transcribed, vm a quaslcontinual representation, into a continuous, non-local in spatial varmbles field theory, whmh Is then quantlzed by the path integral method.
A purpose of this note is to lbrmulate a perturbaUve scattering theory for a faithful lattice version of the 4 + 1 theory. For this we use, as a main technical tool, so called quaslcontlnual representation of latlme values of the field [1~2]. We refer to these papers for all the techmcal and notational quesUons. Let us start with the classmal lagranglan of the theory L = ½OuqSO/a~b ½m2~b2 - @/4!) ~b4 .
(1)
Introducing the eqmdlstant lattices on each axis, with lattice constants a0, a 1 ..... aD we pass to the lattice values of the field
(~(na), na = (n0a 0, n l a l ..... nDaD), n u @ Z} .
(2)
Performing the quaslcontmual interpolation between these values one replaces them with a single field Sa of contmuous vanablesxU (cf. refs. [1,2] and also refs [ 3 - 7 ] ) D
Oa(X) = ~I a u ~n CMn) 8a(X - na) ~ OC(a, 0 ) .
(3)
~=0
An action functional of this field is given by the formulae D
=f dxi½Ouq,a(X) OUg~a(x) _ ~m2q)~(x) l
~Xf
dx 1 dx2 dx3 dx4Ma(Xl,X2,X3,x4J~a(Xl)(Oa(X2JOa(X3J~a(X4)
(4) where 0,~b(n) are the quasicontmual derivatives, and M a are known form factors. From this we infer that the canonical lagrangmn is
_a
I
")2
LalX ) - ~Ojjd~a(X) O#Oa(X) - ~tn-~a(X ) -
X ~.. f dx 1 dx2
dx3 Ma(X,Xl,XZ,X 3) ~)a(Xl )d~a(X2) ~a(X3) ~)a(x)
= LOa(x) + Lint(x) ,
0 0 3 1 0 1 6 3 / 8 3 / 0 0 0 0 - 0 0 0 0 / $ 03.00 © 1983 North-Holland
(5)
275
Volume 122B, number 3,4
PHYSICS LETTERS
10 March 1983
while the effectwe lagranglan IS of the form
f~a(X) ~OtadPa(X)~t~(Pa(X)
~m24~2(x)
dx 2 dx3Ma(X,Xl,X2,x3) (Pa(X1)~a(X2)~)a(X3)~)a(X) .
(6)
The appearance of two lagranglans xs a typical feature of a non-local field theory [8,9]. The last lagranglan determines the equation of motion of the field q~a through the Euler-Lagrange equation
-- m2) (Pa(x) = ~X. Jf d x 1 dx2 dX3Ma(X,Xl,X2,X3)dPa(Xl)dPa(X2)~a(X3)
(7)
The canomcal lagranglan L a serves for constructing a hamdtoman of the field q~a, and thus plays a central role m the quantlzatlon procedure. The quantizatxon is possible ff a theory is local m time. Thus removing a lattice from the rime axis by taking the limit a 0 -+ 0, and setting a 1 = ... = aD = a one gets for the field
Cka(t,x) = a D ~ (p(t, n) 5a(x - na) E QC(a, 0 ) .
(8)
I1
The field is continuous m the txme variable while m the spanal variables the dependence ,s of quas~contmuous character. The form factor M a becomes [1,2]
Ma(X, Xl,X2,x3)
5(xO-xO) 5(xO-xO2)~(xO -xO)Ma(XX ,X ,l2x,3)3
"
,
(9)
aO--~O
and integrations over the time variables are removed m the formulae (5), (6) and (7). According to the general theory of constructing of constants of motion for non-local theories one gets for the hamdtonian the formula [8,9] D 2 k=l [3kq~(x)]2 +~m24~2(x) +
f dDx I dDx2 dDx 3 Ma(X, Xl, x2, x3)~b(Xl)~(x2)q~(x3)~(x)),
(1 0)
where
~o(x)=-~)a(t,x)lt= 0 ,
n(x)---(~.(t,x)lr=o.
(11)
The hamlltoman xs chosen in such a way as to secure the eqmvalence of the canonical Hamdton equatmns with eq. (7). The theory admits the path-integral quantlzation procedure [ 1 0 - 1 2 ] , and yields, after necessary modifications, the following formula for the S-matrix describing a scattenng on the interaction :LaTM. + J " q~, where J(x) is an external classical source
S(J, X) = : e x p { - ¢ 0 [a+, a] K8 ~SJ} Z [J, X] : ,
(12)
and Z [J, X] = N - 1 (X) exp (l .Lmt [8/81J] '}[ao=0 exp (~ 1JAiJ}.
(13)
Here the functmn A(x - - y ) is gwen by the formula oo
A(x - y ) = (2~r)-W+l)
~f dpof -~
dDp exp[_lP0(X0 _ y 0 ) + lp(x - y ) ] [(p0) 2 - (p)2 _ m 2 + le]-1 .
~ *
s4* is a compactlficatlon of the set M of momenta
276
(14)
Volume 122B, number 3,4
PHYSICS LETTERS
10 March 1983
05)
s~ = {p;-rr/a<~Ptc<~rr/a,k = 1..... D} , A =rr/a . The free field qS0[a+, a] (x) as given by Oo[a +, a] (x) = (2rr) -D/2 f dDp {a(p) exp [-l(x0COp _o~* (2C°P)1/2
px)] + h.c.}
(16)
and the symbol Llnt . [ ] stands for the expression :Lmt[qs] =
dx dx dx dx
U
......
Ma(x,x ,x ,x '
"
)lao= 0
"'
X [ ( a ( x ) ( a ( x ' ) O ( x " ) ~ ( x " ' ) - 6 A ( x - x')c)(x")c~(x " ' )
+ 3 A(x
--x') A(x" -- x ' " ) ]
(17)
.
Taking into account the formulae co
q~(x) = (err) -(D+I)/2 f
dP0 f
(:8)
dDp ep(p) e x p ( - i p x )
(slmIlat for J(x)), and
Ma(X,X',X",x'")=(2rr)-3(D+l) f dp dp' dp" dp'"6(p +p' + p " + p " ' ) rood 2A s~Sx.~* X exp[l(px + p'x' + p . .x. . + p ' " x ' " ) ]
(19)
whe re
dl,=dPodPl...dPD ,
(20)
.~* = s ~ X ... X M~) ,
we get for the generating functional of the full Green functions
ZlJ, X l = N - ' ( X ) e x p [ - ~ ( 2 r r ) - ( ° + l ) •
f
( X
dpdp'dp"dp"'6(p+p'+p"+
+ p,) aIJ(p)61J(p'i&J(p")&J(p") (~_J
X exp
p"') rood 2A, a o =
_~ ~ x 9i,,
0
8(p + p')
6p 2 - m 2 +le 61J(p")&J(p") + 3 p 2 - m 2 +lep "2 - m 2 + le ,l-J
lS(q+q') ,j(q,)). dq dq' lJ(q) q2 _ m 2 + le
(e:)
The Feynman roles for constructing diagrams are: I
internal line.
vertex"
p
\>
-
~p,,, ~,p"
p2 _ m 2 + le '
p +p' +p" +
dp loop integration ,~rrD+-- ,
p" = 0 ~ -IX, rood 2A, a0 =0
symmetry factor S .
(22)
\
The Feynman rules in the coordinate space are more comphcated since they involve "generalized vemces" which appear under the form factor Ma(x , x', x", x " ) . 277
Volume 122B, number 3,4
PHYSICS LETTERS
10 March 1983
References [11 [2] [3] [41 [5] [6} [7] 18]
[91 [10] [11] [ 12]
278
W Garczyfiskl and J Stelmach, J Math Phys 22 (1981) 1106. W Garczyfiskl and J. Stelmach, J Math. Phys 23 (1982) 1168 K. Wilson, Phys. Rev. D10 (1974) 2445 S.D Drell, M Wemsteln and S Yanklelowlcz, Phys. Rev D13 (1976) 3342. D. Rogula, Bull Acad. Polon. ScL, s~rle Scl. Techn 13 (1965) 337; see also- Quaslcontmual theory of crystals, Arch Mech 28 (1976) 563. J.A. Krumhansl, Generahzed continuum field representation for lattice vibrations, in Lattice dynamms, ed R.F. Walhs (Pergamon, New York, 1965) p. 627. I A. Kunm, Pnkl. Mat. Mech. 30 (1966) 542; see also Teona upruglkh sred s mlkrostrukturol (Nauka, Moscow, 1975) [m Russian] J. Rzewuskl, Field theory, Part I (Polish Scientific Publishers, Warsaw, 1964) Ch I1. W Garczyfiskl and J. Stelmach, Bull. Acad Polon., C1. IIl 28 (1980) 141. J Rzewuskl, Field theory, Part II (Pohsh Scientific Pubhshers, Warsaw, 1975) L.D Faddeev, Les ttouches Lectures, Session XX (North-Holland, Amsterdam, 1976). D J Amlt, Field theory, the renormaltzatlon group and critical phenomena (McGraw-Hill, New York, 1978).