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Scattering in a quantum field theoretical model in two space-time dimensions
Volume 52B, number 4
PHYSICS LETTERS
28 October 1974
S C A T T E R I N G IN A QUANTUM F I E L D T H E O R E T I C A L M O D E L IN TWO SPACE-TIME DIMENSIONS H.G. DOSCH
lnstitut flir Theoret. Physik, UniversitdtHeidelberg, Germany and V.F. MI]LLER
FachbereiehPhysik, UniversitdtTrier-Kaiserslautern,Germany Received 19 September 1974 It is shown that compositebosons appearing in quantum field theoretical models of the type studied by Mattisand Lieb cannot be produced by collisionsof physicalfermionsor antffermions. Recently field theoretical models in two space-time dimensions received considerable attention by elementary particle physicists. Such models are partly used to describe composite hadrons [1-3] or are directly connected with suitably chosen physical processes with some degrees of freedom being frozen [4]. In this note we exhibit some properties of the S-matrix of a field theoretical model for interacting fermions in two space-time dimensions. Although not Lorentz-invariant, this model incorporates the complications of a relativistic quantum field theory due to its particle-antiparticle symmetry. It has been investigated by Mattis and Lieb [5] with special emphasis on the associated boson f'trst considered by Jordan [6]. Starting with the Fock-representation of the canonical anticommutation relations (CAR) for fermions and antifermions,
[bk, b~]+= [Ck, C~,l.=Skk, ,
k=(2rr/L)n,
nEZ
(1)
aU other anticommutators vanishing, the vacuum vector ~2o satisfying bk~2 o = Ck~2o = O, the canonical fermion field operator at t = 0 in the interval -L/2 < x <<.L/2 is defined:
1 / Ct-k \~ ~(x,=-~{k~oeXp(ikX,(cbtk)+k~
(2,
Introducing for p = (2n/L) n, nE Z the operators L/2 = 1,2 p(~) -= _L/J/2 dx exp (ipx) : ff~(x) ~a(x):,
(3)
one obtains for p 4= 0 the boson-operators
atp--~lL{O(p)p(l'+O(-p)p(2)
},
ap = 21p~L {®(P) P_p (1) + O ( - P ) P (2)) -p ,
(4)
satisfying canonical commutation relations (CCR)
[ap,ap,] = 0;
[ap,atp,] = 6pp,.
(5)
The operator apt is the adjoint of ap, ap[2 o = 0. The momentum operator
445
Volume 52B, number 4
PHYSICS LETTERS
28 October 1974
L/2
P=--i_Lf/2 dx :~bt(X)~x if(x):
(6)
implies that the subscripts of the fermion and boson destruction and creation operators are the momenta of the corresponding bare particles. We investigate scattering caused by the Hamilton-operator [5 ]
L/2 H=--i_Lf/2 dx:~k'(x) o3~x
{ L/2 L/2 1 ) ~k(x):+h _Lf~2 dX_Lf/2 dy:~k'(x)$(x):V(x-y):$'(y)ff(y):-~-/~p lplOp (7)
with a potential
V(x-y) =L1 p~ Up exp ( i p ( x - y ) ) , where p = (2rr/L)n, nE Zand
(8)
Up= O_p =Up. From eqs. (2), (3) and (4) follows:
H= ~tpl{b;bp +ctpCp)+ 2rr ~--~p Iplop ( 2a;ap +a~ap +apa_p}. p
(9)
For Zp p202 < ~ this is a well-defined operator in F. Since the bosons appear as composite particles we ask whether they can be produced by fermion-fermion or fermion-antifermion collisions. Due to to the compositeness of these bosons, a scattering theory is required depending exclusively upon the total Hamilton operator. Such a theory has been deviced by Eckstein [7]. To all particles being observable objects correspond two sets of asymptotic creation and annihilation operators satisfying CCR or CAR, furthermore the commutation relations of these operators with the total Hamilton- and the momentum operator are those of free particles. The boson operators (4), already satisfying CCR, have the commutation relations
[P,a~] =patp,
[H,atp] = lp'(l+ Xhop)apt+ ~-Iplvp X a_p.
(10, 11)
For all ~0pER the new boson operators ~p~---apt cosh ~Op+a_p sinh ~Op,
~p=ap
cosh ~Op+at_p sinh ~Op
(12)
satisfy CCR and are adjoint to each other, moreover [p, atp I = p~tp.
(13)
Enforcing the relation [H, ~ ] =Ep~;
(14)
with Ep real positive, we obtain, if (X/n) Op>
Ep = Ipl(l+~Op) 1/2,
-
~-, t "q p the solution
X -1 tanh 2~pp -x--~vp ( l + ~ V p ) .
(15)
This solution is unique. Thus for vp = p-2 a relativistic boson [5,8] with mass 2X/It appears. The solution (15) obtained using Eckstein's scattering theory agrees with the results of ref. [5], where a unitary transformation has been employed. 446
Volume 52B, number 4
PHYSICS LETTERS
28 October 1974
The uniqueness of the solution o f eq. (14) is in contrast to the general case, where two solutions occur [7], an incoming and an outgoing one. Hence we conclude that for the boson we have
Zip -- Up^-in--Up-~°ut
(16)
implying that these bosons cannot be produced by fermion collisions. Assuming that we found asymptotic fermion (antifermion) operators/~ht#(j~ b'~t#), where the symbol # means creation or annihilation operator, and a unique physical vacuum vector ~2, mapped to zero by all asymptotic annihilation operators, then we can conclude - due to the relations [7] in in [z# ~out#l tUp,jr ol u t # lj -- [~# (17) t p'Jl J = 0, that all production matrixelements vanish: ^
(af, l""apn
~ fint~in÷ [2)=0.
"11 "12'
•
(t~p~1 ...~pn foutt~-outt ~ ~-inj't-in+~a = O, ...
'13
J14
This proves our assertion Finally we consider the limit case, where the interval replacements b #k - ~ b #(k)
'Jtt "12'
(-L/2, L/2)
(18)
J
doesn't have finite length. Then we have the
and c~-~ c#(k) t
where the new creation (annihilation) operators, depending on k E R , act in a Fock-space F' with vacuum ~2o and satisfy
[b(k), b+(k')] + = [c(~), e+(k')] + = ~(g-k').
(19)
Then in the preceding formulae all summations over discrete momenta have to be replaced by integrations and each factor L -1/2 by (2r0 - I / 2 . The boson operators a(p) satisfy
[a(p),a t (p')] = 5(p- p').
(20)
In contradistinction to the discrete case eq. (9) does no longer define an operator in F' but only a hermitean bilinear form on a dense domain. Nevertheless, eqs. (11) and (14) can be considered purely algebraically, to be realized in an appropriate Hilbert-space. Since eqs. (12) and (15) depend on these algebraic properties only, they remain valid. Assuming as before to have found asymptotic fermion and antifermion operators satisfying eqs. (17) we may express H in terms of all asymptotic operators and define a vacuum mapped to zero by the asymptotic annihilation operators. Thus arrived at a representation of the CCR and CAR unitarity inequivalent to the original Fock-representation, all conclusions drawn from eqs. (18) remain valid.
References [ll [2] [3] [4] [5] [6] [7]
A. Casher, J. Kogut and L. Susskind: Tel Aviv preprint TAUP - 373 - 73. C.E. Caxlson, L.N. Chang, F. Mansouri and J.F. WiUemsen, Phys. Lett. 49B (1974) 377. G. 't Hooft, CERN preprint TH 1820 (1974). S. Fubini, C. Rebbi, CERN preprint TH 1809 (1974). D.C. Mattis and E.H. Lieb, J. Math. Phys. 6 (1965) 304. P. Jordan, Z. Physik 93 (1935) 464; 102 (1936) 243; 105 (1937) 114,229. H. Eckstein, Nuovo Cim. IV (1956) 1017; R. Haag, Phys. Rev. 112 (1958) 669; D. Ruelle, Helv. Phys. Acta 35 (1962) 147. [8] J. Sehwinger, Phys. Rev. 128 (1962) 2425; J.H. Lowenstein and J.A. Swieca, Ann. Phys. (N.Y.) 68 (1971) 172. The infinitely rising Coulomb - potential in one dimensional space does however not lead to proper scattering conditions for fermions. Note that in our treatment no use is made of the relativistic energy momentum relation for the bosom 447
PHYSICS LETTERS
28 October 1974
S C A T T E R I N G IN A QUANTUM F I E L D T H E O R E T I C A L M O D E L IN TWO SPACE-TIME DIMENSIONS H.G. DOSCH
lnstitut flir Theoret. Physik, UniversitdtHeidelberg, Germany and V.F. MI]LLER
FachbereiehPhysik, UniversitdtTrier-Kaiserslautern,Germany Received 19 September 1974 It is shown that compositebosons appearing in quantum field theoretical models of the type studied by Mattisand Lieb cannot be produced by collisionsof physicalfermionsor antffermions. Recently field theoretical models in two space-time dimensions received considerable attention by elementary particle physicists. Such models are partly used to describe composite hadrons [1-3] or are directly connected with suitably chosen physical processes with some degrees of freedom being frozen [4]. In this note we exhibit some properties of the S-matrix of a field theoretical model for interacting fermions in two space-time dimensions. Although not Lorentz-invariant, this model incorporates the complications of a relativistic quantum field theory due to its particle-antiparticle symmetry. It has been investigated by Mattis and Lieb [5] with special emphasis on the associated boson f'trst considered by Jordan [6]. Starting with the Fock-representation of the canonical anticommutation relations (CAR) for fermions and antifermions,
[bk, b~]+= [Ck, C~,l.=Skk, ,
k=(2rr/L)n,
nEZ
(1)
aU other anticommutators vanishing, the vacuum vector ~2o satisfying bk~2 o = Ck~2o = O, the canonical fermion field operator at t = 0 in the interval -L/2 < x <<.L/2 is defined:
1 / Ct-k \~ ~(x,=-~{k~oeXp(ikX,(cbtk)+k~
(2,
Introducing for p = (2n/L) n, nE Z the operators L/2 = 1,2 p(~) -= _L/J/2 dx exp (ipx) : ff~(x) ~a(x):,
(3)
one obtains for p 4= 0 the boson-operators
atp--~lL{O(p)p(l'+O(-p)p(2)
},
ap = 21p~L {®(P) P_p (1) + O ( - P ) P (2)) -p ,
(4)
satisfying canonical commutation relations (CCR)
[ap,ap,] = 0;
[ap,atp,] = 6pp,.
(5)
The operator apt is the adjoint of ap, ap[2 o = 0. The momentum operator
445
Volume 52B, number 4
PHYSICS LETTERS
28 October 1974
L/2
P=--i_Lf/2 dx :~bt(X)~x if(x):
(6)
implies that the subscripts of the fermion and boson destruction and creation operators are the momenta of the corresponding bare particles. We investigate scattering caused by the Hamilton-operator [5 ]
L/2 H=--i_Lf/2 dx:~k'(x) o3~x
{ L/2 L/2 1 ) ~k(x):+h _Lf~2 dX_Lf/2 dy:~k'(x)$(x):V(x-y):$'(y)ff(y):-~-/~p lplOp (7)
with a potential
V(x-y) =L1 p~ Up exp ( i p ( x - y ) ) , where p = (2rr/L)n, nE Zand
(8)
Up= O_p =Up. From eqs. (2), (3) and (4) follows:
H= ~tpl{b;bp +ctpCp)+ 2rr ~--~p Iplop ( 2a;ap +a~ap +apa_p}. p
(9)
For Zp p202 < ~ this is a well-defined operator in F. Since the bosons appear as composite particles we ask whether they can be produced by fermion-fermion or fermion-antifermion collisions. Due to to the compositeness of these bosons, a scattering theory is required depending exclusively upon the total Hamilton operator. Such a theory has been deviced by Eckstein [7]. To all particles being observable objects correspond two sets of asymptotic creation and annihilation operators satisfying CCR or CAR, furthermore the commutation relations of these operators with the total Hamilton- and the momentum operator are those of free particles. The boson operators (4), already satisfying CCR, have the commutation relations
[P,a~] =patp,
[H,atp] = lp'(l+ Xhop)apt+ ~-Iplvp X a_p.
(10, 11)
For all ~0pER the new boson operators ~p~---apt cosh ~Op+a_p sinh ~Op,
~p=ap
cosh ~Op+at_p sinh ~Op
(12)
satisfy CCR and are adjoint to each other, moreover [p, atp I = p~tp.
(13)
Enforcing the relation [H, ~ ] =Ep~;
(14)
with Ep real positive, we obtain, if (X/n) Op>
Ep = Ipl(l+~Op) 1/2,
-
~-, t "q p the solution
X -1 tanh 2~pp -x--~vp ( l + ~ V p ) .
(15)
This solution is unique. Thus for vp = p-2 a relativistic boson [5,8] with mass 2X/It appears. The solution (15) obtained using Eckstein's scattering theory agrees with the results of ref. [5], where a unitary transformation has been employed. 446
Volume 52B, number 4
PHYSICS LETTERS
28 October 1974
The uniqueness of the solution o f eq. (14) is in contrast to the general case, where two solutions occur [7], an incoming and an outgoing one. Hence we conclude that for the boson we have
Zip -- Up^-in--Up-~°ut
(16)
implying that these bosons cannot be produced by fermion collisions. Assuming that we found asymptotic fermion (antifermion) operators/~ht#(j~ b'~t#), where the symbol # means creation or annihilation operator, and a unique physical vacuum vector ~2, mapped to zero by all asymptotic annihilation operators, then we can conclude - due to the relations [7] in in [z# ~out#l tUp,jr ol u t # lj -- [~# (17) t p'Jl J = 0, that all production matrixelements vanish: ^
(af, l""apn
~ fint~in÷ [2)=0.
"11 "12'
•
(t~p~1 ...~pn foutt~-outt ~ ~-inj't-in+~a = O, ...
'13
J14
This proves our assertion Finally we consider the limit case, where the interval replacements b #k - ~ b #(k)
'Jtt "12'
(-L/2, L/2)
(18)
J
doesn't have finite length. Then we have the
and c~-~ c#(k) t
where the new creation (annihilation) operators, depending on k E R , act in a Fock-space F' with vacuum ~2o and satisfy
[b(k), b+(k')] + = [c(~), e+(k')] + = ~(g-k').
(19)
Then in the preceding formulae all summations over discrete momenta have to be replaced by integrations and each factor L -1/2 by (2r0 - I / 2 . The boson operators a(p) satisfy
[a(p),a t (p')] = 5(p- p').
(20)
In contradistinction to the discrete case eq. (9) does no longer define an operator in F' but only a hermitean bilinear form on a dense domain. Nevertheless, eqs. (11) and (14) can be considered purely algebraically, to be realized in an appropriate Hilbert-space. Since eqs. (12) and (15) depend on these algebraic properties only, they remain valid. Assuming as before to have found asymptotic fermion and antifermion operators satisfying eqs. (17) we may express H in terms of all asymptotic operators and define a vacuum mapped to zero by the asymptotic annihilation operators. Thus arrived at a representation of the CCR and CAR unitarity inequivalent to the original Fock-representation, all conclusions drawn from eqs. (18) remain valid.
References [ll [2] [3] [4] [5] [6] [7]
A. Casher, J. Kogut and L. Susskind: Tel Aviv preprint TAUP - 373 - 73. C.E. Caxlson, L.N. Chang, F. Mansouri and J.F. WiUemsen, Phys. Lett. 49B (1974) 377. G. 't Hooft, CERN preprint TH 1820 (1974). S. Fubini, C. Rebbi, CERN preprint TH 1809 (1974). D.C. Mattis and E.H. Lieb, J. Math. Phys. 6 (1965) 304. P. Jordan, Z. Physik 93 (1935) 464; 102 (1936) 243; 105 (1937) 114,229. H. Eckstein, Nuovo Cim. IV (1956) 1017; R. Haag, Phys. Rev. 112 (1958) 669; D. Ruelle, Helv. Phys. Acta 35 (1962) 147. [8] J. Sehwinger, Phys. Rev. 128 (1962) 2425; J.H. Lowenstein and J.A. Swieca, Ann. Phys. (N.Y.) 68 (1971) 172. The infinitely rising Coulomb - potential in one dimensional space does however not lead to proper scattering conditions for fermions. Note that in our treatment no use is made of the relativistic energy momentum relation for the bosom 447