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Hilbert space sectors, dynamical charges and spontaneous symmetry breaking in classical field theory
Volume 62B, number 1
PHYSICS LETTERS
10 May 1976
HILBERT SPACE SECTORS, DYNAMICAL CHARGES AND SPONTANEOUS
SYMMETRY BREAKING IN CLASSICAL FIELD THEORY C. PARENTI
Istituto Matemattco "S. Pincherle ", Umversi O, o f Bologna, Bologna, Italy
F. STROCCHI Scuola Normale Superiore and LN.EN., Sez. di Pisa, Pisa, Italy
and G. VELO Istituto di Fisica ",4. Rtghi", University of Bologna and 1.N.I~:N., Sez. di Bologna, Bologna, Italy
Received 20 February 1976 Results concerning existence and properties of Hilbert space sectors in the fandly of solutions of classical non-linear field equations are presented. Within this framework the existence of conserved dynamical charges and the occurrence of spontaneous symmetry breaking is discussed.
It has recently become clear that the study of solutions of classical non-linear field equations may provide a way of understanding basic propemes of elementary particles [1 ]. The quantum field theory is constructed by considering some simple stable solutions of the field equations (such as the constants which minimize the energy [2] and the solitons [3]) and analyzing small perturbations around them. This procedure is substantially different from the Fock representation approach, which is based on the solutions of the free equations, and involves the non-linear character of the equations in a much more radical way. A systematic investigation of the structure of solutions of classical non-linear equations seems to be lackmg in the literature. The purpose of this note is to present some rigorous results along the above lines and in particular to prove the existence of charges which have dynamical rather than group theoretical origin, the existence of Hilbert space sectors, their stability under time translation and the occurrence of spontaneous symmetry breaking. For the sake of simplicity we will mostly discuss the case of a one component field, under suitable and simple assumptions on the potential. Generalizations to n-component fields and to a larger class of potentials as well as detailed proofs will appear elsewhere.
More specifically we will analyze the solutions of the following Cauchy problem in s space dimensions (1 ~
a~
(1)
~(x,0) = G ( x ) , ~a q ~( x 0, ) = % ( x ) wlfich we rewrite in the form of an integral equation:
¢ct)!
o
l=s -i~(~(=))]
where W(t) is the one parameter group generated by (AOl0 )" The potential U(z) 1s assumed throughout the whole note to satisfy the following assumptions: i) There exists two non-negative constants e,/3 for which U(z) >1 - ~ - (32 for all z E R 1 , d) s = 1 U(z) is any entire function, s = 2 U(z) = Nnan zn is any entire function for which Nnnn/21an I Izl n < oo for all z E R 1 , s = 3 U(z) is any twice differentlable function for which [ d'(z)l ~< const(1 + z 2) for all z E R 1 . In a preceding paper [4] it has been proved that 83
Volume 62B, number 1
PHYSICS LETTERS
for any mitlal data ( ,(~o ) in the space X of real func~o tions with locally finite kinetic energy the mtegral equation (2) has a unique X-valued solution r¢(t)~ ~¢ (t)J continuous in the time variable. In the following we will always refer to such type of solutions. The precise 2 R s) definition of X is X = H~oc(RS ) e~Lloc( (~bEH~oc(aS), ~EL2oc(RS))which means that f~z I(V~b)2 + ~b2 + $21 dx < oo for all bounded open sets ~2 of R s. A very interestmg structure emerges when one looks for a physical Interpretation of the theory. Since initml data, for which the energy difference of the corresponding solutions is mfinite, cannot be realized in the same "physical world", the set ~ of solutions gets naturally divided into classes, two solutions belonging to the same class if they have relatively firote energy. Thas suggests to group together solutions which may be regarded "small" perturbations one of the other. More precisely ( ~ ' ~ ) will be considered as .. "~ 0 if: a . .small perturbation of(¢ft))
-=
\~(t)]
t wlth values in
(3)
k~/(t)- t~(t) HI(RS)*L2(RS).
(the space H 1 (R s) * L2(R s) is the space of the X's and ~"s for which fRsI(VX) 2 + X2 + ~'21 dx < ~). The above relation among pair of solutions of eq. (2) is clearly an equivalence relation and induces a partition of ~r into classes. As a further physical requirement we consider only those classes which are left invariant under time translation. This amounts to considering only those solutions for which the time evolution is a "small" perturbation of the initial data in the sense that: (~(t) - ¢(0) 1
is a continuous function of t with values m
\$(t)
HI(RS)* L2(RS).
¢(0)]
(4)
These two "smallness" conditions allow us to introduce Hilbert space sectors in the set X of initial data. two elements of X (~), (f,), whose corresponding solutions satisfy condition eq. (4), are said to belong to t h e s a m e H i l b e r t s e c t o r i f ( ' , .(t) . ) is a sm a 11 pe rtur b atlon (t) of ( ~ ) i n the sense of relation eq. (3) (~b(0) = ~b, $(0)~'~b, $'(0) = $', ~b'(0) = $'). Each Halbert sector is uniquely determined by any of its elements (~) and will be denoted by H(¢,,¢ ). 84
10 May 1976
The following two theorems are crucial for the existence and construction of Hilbert sectors. Theorem 1. If a solution (~ttl) of eq. (2)satisfies condition eq. (4) and $oEL°°(RS) (~b(0) = ~o, ~(0) = $o), then'
/
aU A~bo - --~(~bo) E H -1 (R s)
(¢oEL2(RS).
Theorem 2. If (~) E X satisfies the following conditions: OU (~b)E H - 1 ( R s) i) A~b -- ~-~ ii) S E L 2 ( R s) ill) ~bEL~(RO then (~) determines a Hilbert sector, whose elements are all the (~',) such that q~' - ~bEH1 (R s) and ~ ' EL2(RS). We recall that f(x)ELOO(R s) if f (x) is a measurable essentially bounded function on R s and that .f(x)EH -1 (R s) if f is a tempered distribution with flY(k)[2/(1 + k 2) dk < oo. It is worthwhile to note that condition i) of Theorem 2 is satisfied by the static solutions of eq. (2). In particular, the elements of X of the form ~¢=0 r~ = const j determme Hilbert sectors if they extremize the potential (Goldstone theorem [2] ): (~U/aO) (const) -- 0. Particularly interesting is the structure of those Hilbert space sectors whose elements (9) have a good "ultraviolet behaviour , i.e., V ~bEL2(RS). For these sectors one may easily define for all (f,) EH(~,,¢) a linear momentum functional:
P(,,,)(,', ,,) = f ( , ' v , ' -
, r e ) dx
Rs and a total energy functional: '
'
"
([-dv~)2+~v
+u(~')]
Rs -- [{-(V~b)2 + ~- ~2 + U(~b)] ) ~ o ( ~ - )
dx
where ~OECo(RS) with co = 1 in a neighbourhood of
Volume 62B, number 1
PItYS1CS LETTERS
the origin, and xoERS (the hmit does not depend on the choice of co and Of Xo). This structure of Hfibert space sectors allows us to Introduce as an invanant of each sector the "behaviour at infinity" of its elements. In fact ff (~',)EH(~, qj), qS' - ~ tends to zero at oo in almost all directions as shown by the followmg techmcal Theorem 3. Let q~EH 1 (R s) and define (tlansttlon to polar coordinates): ~(r, ~2)-= O(x),
x = r[2, r > 0,
~2ES s - 1 .
(5)
Then for almost all [2ES s - 1 (the unit sphere in R s) one has. llm r (s-l)~2 ~p(r, f2) = O. r-*+~
Consequently ff ~ has a finite limxt a(~2) at oo in almost all directions, q~' has the same limit. The function a([2) identifies a "charge" which is constant in time and it is the same for all elements of the sector. For s ~> 3 the existence of the limit a([2) is guaranteed for all those sectors H(O,~) for which VOEL2(RS), as shown by the following techmcal Theorem 4. Let s 1> 3, ¢EH~oc(RS ) with VC~EL2(RS). Then for almost all YZESs - 1 the function ~(r, [2) (see eq. (5)) has a finite hmit ~(oo, [2) as r-+ + oo and: lim r (s- 2)/2 (~(r, [2) - ~(oo, [2)) = 0. r---> + ~
The idea of associating a charge to the asymptotic behaviour of non-hnear evolution equations has been first discussed by Finkelstein and Mlsner [5]. The framework constructed so far permits a rigorous treatment of this concept. It is important to stress that the existence of non trivial charges is strictly'related to the structure and existence of the Hllbert space sectors, i.e., to the dynamics o f the theory (dynamical charges). These charges do not arise as generators of symmetry groups as the ordinary charges m quantum field theory. Finally it is possible to understand on a rigorous level the occurrence of spontaneous symmetry breakrag. For this purpose we consider a theory with multicomponent fields
n
n
10 May 1976
to which the results previously &scussed can be applied. We can define a local internal symmetry as a map
Tg \~(X)]
~Jg(O(x)) ~(x)]
Induced by an mverhble and regular transformation
g : R n -> R n with Jacohian matrix denoted by Jg, which has the property of carrying solutions of eq. (2) into solutions of the same equation. It turns out that, under mild assumptions on the potential U, the set of local internal symmetries Is precisely the set o f all affine transfolmations z -->A z + a such that A T A = ~,rtRn and U(Az + a) = 3,U(z) + U(a) (we suppose here U(z) not ldenhcally zero and U(O) = 0). One may see that in general the Hilbert space sectors are not necessarily preserved by local internal symmetry. We will say that a (local Internal) symmetry Tg Is spontaneously broken in the sector H(~, qj) if it does not map this sector into itself. It is useful to remark that local internal symmetries which are not spontaneously broken in a given sector can be realized there by umtary operators.
References [1] T.D. Lee and G.C. Wick, Phys. Rev. D9 (1974) 2291, B. Ferretn, Art1 Acc. Scienze Bologna, Sene XII, Tomo 1 (1974) p. 45; G. 't ttooft, Nucl. Phys. B79 (1974) 276; B. Ferretti and G Velo, Nuovo Clmento Lett. 10 (1974) 451; R. Dashen, B Hasslacher and A. Neveu, Phys Rev. Dl 0 (1974) 4114, 4130, 4138, J. Goldstone and R Jacklw, Phys. Rev. D l l (1975) 1486; S. Coleman, Phys Rev. D l l (1975) 2088, J. FrBlich, Phys. Rev Lett. 34 (1975) 833; J.L. Gervais and R. Saklta, Phys. Rev DI 1 (1975) 2943; S. Mandelstam, Phys. Rev. D11 (1975) 3026, L.D. Fadeev, IAS lectures and references thereto, April 1975, N. Christ and T.D. Lee, Phys. Rev. D12 (1975) 1606 [2] J. Goldstone, Nuovo Clmento 19 (1961) 154. [3] G.B. Whltham, Linear and nonhnear waves (J. Wiley, New York, 1974). [4] C. Parenn, F. Strocchl and G. Velo, Phys. l_.ett 59B (1975) 157 and preprmt S.N S. 22/1975, to appear m Ann. Scuola Norm. Sup. (hsa). [5] D Flnkelstem and C.W. Mlsner, Ann Phys. 6 (1959) 230; D. Flnkelsteln, J. Math. Phys. 7 (1966) 1218.
85
PHYSICS LETTERS
10 May 1976
HILBERT SPACE SECTORS, DYNAMICAL CHARGES AND SPONTANEOUS
SYMMETRY BREAKING IN CLASSICAL FIELD THEORY C. PARENTI
Istituto Matemattco "S. Pincherle ", Umversi O, o f Bologna, Bologna, Italy
F. STROCCHI Scuola Normale Superiore and LN.EN., Sez. di Pisa, Pisa, Italy
and G. VELO Istituto di Fisica ",4. Rtghi", University of Bologna and 1.N.I~:N., Sez. di Bologna, Bologna, Italy
Received 20 February 1976 Results concerning existence and properties of Hilbert space sectors in the fandly of solutions of classical non-linear field equations are presented. Within this framework the existence of conserved dynamical charges and the occurrence of spontaneous symmetry breaking is discussed.
It has recently become clear that the study of solutions of classical non-linear field equations may provide a way of understanding basic propemes of elementary particles [1 ]. The quantum field theory is constructed by considering some simple stable solutions of the field equations (such as the constants which minimize the energy [2] and the solitons [3]) and analyzing small perturbations around them. This procedure is substantially different from the Fock representation approach, which is based on the solutions of the free equations, and involves the non-linear character of the equations in a much more radical way. A systematic investigation of the structure of solutions of classical non-linear equations seems to be lackmg in the literature. The purpose of this note is to present some rigorous results along the above lines and in particular to prove the existence of charges which have dynamical rather than group theoretical origin, the existence of Hilbert space sectors, their stability under time translation and the occurrence of spontaneous symmetry breaking. For the sake of simplicity we will mostly discuss the case of a one component field, under suitable and simple assumptions on the potential. Generalizations to n-component fields and to a larger class of potentials as well as detailed proofs will appear elsewhere.
More specifically we will analyze the solutions of the following Cauchy problem in s space dimensions (1 ~
a~
(1)
~(x,0) = G ( x ) , ~a q ~( x 0, ) = % ( x ) wlfich we rewrite in the form of an integral equation:
¢ct)!
o
l=s -i~(~(=))]
where W(t) is the one parameter group generated by (AOl0 )" The potential U(z) 1s assumed throughout the whole note to satisfy the following assumptions: i) There exists two non-negative constants e,/3 for which U(z) >1 - ~ - (32 for all z E R 1 , d) s = 1 U(z) is any entire function, s = 2 U(z) = Nnan zn is any entire function for which Nnnn/21an I Izl n < oo for all z E R 1 , s = 3 U(z) is any twice differentlable function for which [ d'(z)l ~< const(1 + z 2) for all z E R 1 . In a preceding paper [4] it has been proved that 83
Volume 62B, number 1
PHYSICS LETTERS
for any mitlal data ( ,(~o ) in the space X of real func~o tions with locally finite kinetic energy the mtegral equation (2) has a unique X-valued solution r¢(t)~ ~¢ (t)J continuous in the time variable. In the following we will always refer to such type of solutions. The precise 2 R s) definition of X is X = H~oc(RS ) e~Lloc( (~bEH~oc(aS), ~EL2oc(RS))which means that f~z I(V~b)2 + ~b2 + $21 dx < oo for all bounded open sets ~2 of R s. A very interestmg structure emerges when one looks for a physical Interpretation of the theory. Since initml data, for which the energy difference of the corresponding solutions is mfinite, cannot be realized in the same "physical world", the set ~ of solutions gets naturally divided into classes, two solutions belonging to the same class if they have relatively firote energy. Thas suggests to group together solutions which may be regarded "small" perturbations one of the other. More precisely ( ~ ' ~ ) will be considered as .. "~ 0 if: a . .small perturbation of(¢ft))
-=
\~(t)]
t wlth values in
(3)
k~/(t)- t~(t) HI(RS)*L2(RS).
(the space H 1 (R s) * L2(R s) is the space of the X's and ~"s for which fRsI(VX) 2 + X2 + ~'21 dx < ~). The above relation among pair of solutions of eq. (2) is clearly an equivalence relation and induces a partition of ~r into classes. As a further physical requirement we consider only those classes which are left invariant under time translation. This amounts to considering only those solutions for which the time evolution is a "small" perturbation of the initial data in the sense that: (~(t) - ¢(0) 1
is a continuous function of t with values m
\$(t)
HI(RS)* L2(RS).
¢(0)]
(4)
These two "smallness" conditions allow us to introduce Hilbert space sectors in the set X of initial data. two elements of X (~), (f,), whose corresponding solutions satisfy condition eq. (4), are said to belong to t h e s a m e H i l b e r t s e c t o r i f ( ' , .(t) . ) is a sm a 11 pe rtur b atlon (t) of ( ~ ) i n the sense of relation eq. (3) (~b(0) = ~b, $(0)~'~b, $'(0) = $', ~b'(0) = $'). Each Halbert sector is uniquely determined by any of its elements (~) and will be denoted by H(¢,,¢ ). 84
10 May 1976
The following two theorems are crucial for the existence and construction of Hilbert sectors. Theorem 1. If a solution (~ttl) of eq. (2)satisfies condition eq. (4) and $oEL°°(RS) (~b(0) = ~o, ~(0) = $o), then'
/
aU A~bo - --~(~bo) E H -1 (R s)
(¢oEL2(RS).
Theorem 2. If (~) E X satisfies the following conditions: OU (~b)E H - 1 ( R s) i) A~b -- ~-~ ii) S E L 2 ( R s) ill) ~bEL~(RO then (~) determines a Hilbert sector, whose elements are all the (~',) such that q~' - ~bEH1 (R s) and ~ ' EL2(RS). We recall that f(x)ELOO(R s) if f (x) is a measurable essentially bounded function on R s and that .f(x)EH -1 (R s) if f is a tempered distribution with flY(k)[2/(1 + k 2) dk < oo. It is worthwhile to note that condition i) of Theorem 2 is satisfied by the static solutions of eq. (2). In particular, the elements of X of the form ~¢=0 r~ = const j determme Hilbert sectors if they extremize the potential (Goldstone theorem [2] ): (~U/aO) (const) -- 0. Particularly interesting is the structure of those Hilbert space sectors whose elements (9) have a good "ultraviolet behaviour , i.e., V ~bEL2(RS). For these sectors one may easily define for all (f,) EH(~,,¢) a linear momentum functional:
P(,,,)(,', ,,) = f ( , ' v , ' -
, r e ) dx
Rs and a total energy functional: '
'
"
([-dv~)2+~v
+u(~')]
Rs -- [{-(V~b)2 + ~- ~2 + U(~b)] ) ~ o ( ~ - )
dx
where ~OECo(RS) with co = 1 in a neighbourhood of
Volume 62B, number 1
PItYS1CS LETTERS
the origin, and xoERS (the hmit does not depend on the choice of co and Of Xo). This structure of Hfibert space sectors allows us to Introduce as an invanant of each sector the "behaviour at infinity" of its elements. In fact ff (~',)EH(~, qj), qS' - ~ tends to zero at oo in almost all directions as shown by the followmg techmcal Theorem 3. Let q~EH 1 (R s) and define (tlansttlon to polar coordinates): ~(r, ~2)-= O(x),
x = r[2, r > 0,
~2ES s - 1 .
(5)
Then for almost all [2ES s - 1 (the unit sphere in R s) one has. llm r (s-l)~2 ~p(r, f2) = O. r-*+~
Consequently ff ~ has a finite limxt a(~2) at oo in almost all directions, q~' has the same limit. The function a([2) identifies a "charge" which is constant in time and it is the same for all elements of the sector. For s ~> 3 the existence of the limit a([2) is guaranteed for all those sectors H(O,~) for which VOEL2(RS), as shown by the following techmcal Theorem 4. Let s 1> 3, ¢EH~oc(RS ) with VC~EL2(RS). Then for almost all YZESs - 1 the function ~(r, [2) (see eq. (5)) has a finite hmit ~(oo, [2) as r-+ + oo and: lim r (s- 2)/2 (~(r, [2) - ~(oo, [2)) = 0. r---> + ~
The idea of associating a charge to the asymptotic behaviour of non-hnear evolution equations has been first discussed by Finkelstein and Mlsner [5]. The framework constructed so far permits a rigorous treatment of this concept. It is important to stress that the existence of non trivial charges is strictly'related to the structure and existence of the Hllbert space sectors, i.e., to the dynamics o f the theory (dynamical charges). These charges do not arise as generators of symmetry groups as the ordinary charges m quantum field theory. Finally it is possible to understand on a rigorous level the occurrence of spontaneous symmetry breakrag. For this purpose we consider a theory with multicomponent fields
n
n
10 May 1976
to which the results previously &scussed can be applied. We can define a local internal symmetry as a map
Tg \~(X)]
~Jg(O(x)) ~(x)]
Induced by an mverhble and regular transformation
g : R n -> R n with Jacohian matrix denoted by Jg, which has the property of carrying solutions of eq. (2) into solutions of the same equation. It turns out that, under mild assumptions on the potential U, the set of local internal symmetries Is precisely the set o f all affine transfolmations z -->A z + a such that A T A = ~,rtRn and U(Az + a) = 3,U(z) + U(a) (we suppose here U(z) not ldenhcally zero and U(O) = 0). One may see that in general the Hilbert space sectors are not necessarily preserved by local internal symmetry. We will say that a (local Internal) symmetry Tg Is spontaneously broken in the sector H(~, qj) if it does not map this sector into itself. It is useful to remark that local internal symmetries which are not spontaneously broken in a given sector can be realized there by umtary operators.
References [1] T.D. Lee and G.C. Wick, Phys. Rev. D9 (1974) 2291, B. Ferretn, Art1 Acc. Scienze Bologna, Sene XII, Tomo 1 (1974) p. 45; G. 't ttooft, Nucl. Phys. B79 (1974) 276; B. Ferretti and G Velo, Nuovo Clmento Lett. 10 (1974) 451; R. Dashen, B Hasslacher and A. Neveu, Phys Rev. Dl 0 (1974) 4114, 4130, 4138, J. Goldstone and R Jacklw, Phys. Rev. D l l (1975) 1486; S. Coleman, Phys Rev. D l l (1975) 2088, J. FrBlich, Phys. Rev Lett. 34 (1975) 833; J.L. Gervais and R. Saklta, Phys. Rev DI 1 (1975) 2943; S. Mandelstam, Phys. Rev. D11 (1975) 3026, L.D. Fadeev, IAS lectures and references thereto, April 1975, N. Christ and T.D. Lee, Phys. Rev. D12 (1975) 1606 [2] J. Goldstone, Nuovo Clmento 19 (1961) 154. [3] G.B. Whltham, Linear and nonhnear waves (J. Wiley, New York, 1974). [4] C. Parenn, F. Strocchl and G. Velo, Phys. l_.ett 59B (1975) 157 and preprmt S.N S. 22/1975, to appear m Ann. Scuola Norm. Sup. (hsa). [5] D Flnkelstem and C.W. Mlsner, Ann Phys. 6 (1959) 230; D. Flnkelsteln, J. Math. Phys. 7 (1966) 1218.
85