Volume 70B, number 3 PHYSICS LETTERS 10 October 1977

R E L A T I O N B E T W E E N P R O J E C T I V E G E O M E T R Y A N D G R O U P C O N T R A C T I O N

IN S P O N T A N E O U S L Y B R O K E N S Y M M E T R Y T H E O R I E S

C. De CONCINI lstituto di Matematica, Universith di Pisa, Italia

and

G. VITIELLO Istituto di Fisica, Universit~ di Salerno, Italia

Received 19 July 1977

By using projective geometry arguments, it is shown that in a spontaneously broken symmetry theory the linear- ization procedure by which one passes from non-linear Heisenberg fields equations to the linear free fields equations leads to a group contraction mechanism: the symmetry group that appears in observations is a group contraction of the dynamical invariance group.

In quantum field theory the notion of the invariance under certain symmetry transformation can be intro- duced as the invariance of the Heisenberg fields equa- tions under such a transformation of the Heisenbe'rg fields. The dynamics of a physical system is described by these basic equations with given boundary condi- tions. To find a solution of the dynamics in terms of free in- (out-) fields means to carry out a linearization procedure by which one passes from the non-linear Heisenberg equations to the linear free fields equations. In other words, a solution of our theory is found when a mapping between Heisenberg fields ¢(x) and free in- (out-) fields 4~(x) is found:

(al ~(x)lb) = (alF(~(x))lb). (1)

Here la) and jb) denote states of the Fock space for physical particles. Due to the non-linear nature of the dynamical Heisenberg equations, the mapping (1) is expected to be non-linear. As a consequence one na- turally expects to observe different symmetry struc- tures at the dynamical and the phenomenological level of the same original invariance of the theory. Suppose, for example, that the Heisenberg field equations

A(~)~,(x) = J(~,(x)) (2)

are form-invariant under certain internal continuous transformations group G, i.e. are invariant under

d/(x) -+ ~ ' ( x ) = g ( ~ ( x ) ) , g e G, (3)

In eq. (2), A(3) is assumed to be a (n × n)- matrix, if(x) and J are n-rows one-column matrices. The free in-fields ¢(x) will then transform under a transformations group Gin

¢(x) ~ ¢ ' ( x ) = h(q~(x)), h e G in, (4)

such that, from eq. (1),

(a[g(~(x))lb) = (alF(h(~)(x)))b). (5)

The original invariance of the theory requires the in- variance under G in transformations of the free in-field equations and of the S-matrix. Due to the non-linearity of the dynamical map eq. (1), i.e. of the dynamics eq. (2), it is not surprising that G in can be different from G. When this happens we say that dynamical re- arrangement of symmetry occurs [1]. Spontaneously broken symmetry theories present well known examples of dynamical rearrangement of symmetry; it is indeed characteristic of such theories the replacement of the basic symmetry by a different symmetry at the observa- tional level, which manifests itself in the observable ordered states of the physical system. The meaning of the dynamical rearrangement of symmetry is that a basic invariance of the theory cannot disappear but should be always present, although it could be manifest in a different form at the level of physical fields. The re- arrangement of symmetry occurs in the linearization procedure by which one passes from non-linear Heisenberg

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equations to the linear free fields equations, i.e. one passes from a non-linear to a linear realization of the basic invariance of the theory. In ref. [2] it has been analyzed the question about the possible group struc- ture that can appear !n observations, once the invari- ance group at the dynamical level is specified. There, it has been shown that, whenever dynamical rearrange- ment of symmetry occurs, the symmetry group that appears in observations is a group contraction [3] of the dynamical invariance group. The aim of the pres- ent note is to show the occurrence of a group contrac- tion mechanism in spontaneously broken symmetry theories by using projective geometry arguments. In ref. [2] our arguments were mainly based on the sym- metry algebras. In this note our statements involve groups themself.

Suppose that linearization of the Heisenberg equa- tions (2) leads us to the linear homogeneous equations for the free in- (out-) fields 4(x )

K ( b ~ ( x ) = 0. (6)

Linearity of eq. (6) implies that if 41 and 42 are solu- tions, any linear combination of 41 and 42 is a solu- tion. Then, i f 4 is a solution, we can say that the full line passing for 4 and 4 = 0 ( - the origin) represents a solution of eq. (6). In eq. (6) the differential operator K(3) is a (n × n)-matrix. The solutions of eq. (6) form a n-dimensional linear space V n (a realization of which is the Fock sp, ace of physical states, as la> and Ib) in eq.. (1)). Let p n - 1 be the (n - 1)-dimensional projec- tive space of V n, i.e. the space whose elements are the lines for the origin of V n. The tilde denotes an equiva- lence relation in V n , by which two elements 4 and 4' of V n are equivalent if a scalar a exists such that 4 = a4 ' ; clearly 4 and 4' are on the same line for the origin of V n . Note also that the null element 4 = 0 (the origin) of V n has been excluded in our construction. We can sa2/that the elements of the projective s p.ace ~ 'n - 1 are the solutions of eq. (6). Coordinates of P n - 1 are given in the following way: let {4-'}, i = 1, 2 . . . . . n, be coord- inates for V n , A point p of ~ n - 1 is a line for the ori- gin of V n. Let {4}}, i = 1, 2 .... , n, be a point, different from the origin, belonging to the line p. Then {4}}, i = 1,2 . . . . . n, is a point of p, different from the origin, if {4i} = a{4j) , i = i, 2 . . . . . n, with certain scalar a. We see then that p n - 1 has homogeneous coordinates {4i}, i = 1,2 ..... n, defined up to a scalar factor a and with {4i = 0}, for any i, excluded.

Let us consider now the transformations group G under which the Heisenberg equations (2) are invariant. Our task is to study how the group G is affected by the linearization procedure by which one passes to the free field equations (6) in a spontaneously broken symmetry theory (we do not consider here the cases where long- range forces are present; Abelian gauge theories are studied in ref. [4] and the present results apply to this case too. We conjecture the same is true also for non- Abelian gauge theories).

It is well known that the vacuum is not invariant under a symmetry transformation which is spontan- eously broken. Let

(01~ n 10) = v ~ 0 (7)

be the spontaneous breakdown of symmetry condition. Without loss of generality we can assume that there is an asymptotic in- (out-) field 4n corresponding to ~n" To be the physical vacuum no t invariant under spon- taneously broken symmetry transformations of G, there must be at least one of the fields 4i, with i :~ n due to the breaking condition (7), which transforms inhomo- geneously [1,5]:

r

4 i ~ 4 i = 4 i + c i , i--/=n, (8)

where c i is a c-number. Actually, spontaneous break- down of symmetry implies that the fields 4i transform- ing inhomogeneously must form an irreducible repre- sentation of the symmetry group G [1,5]. Assume here for simplicity that they are n - 1 : 4i, i = 1 , 2 . . . . . n - 1. Invariance of the theory requires that the free fields equations for 4i, i =~ n, must be invariant under the in- homogeneous transformations (8). This means that 4i, i :~ n, must be massless fields (the Goldstone fields [6]). On the other hand, the transformations under which the physical vacuum is invariant are linear homogeneous transformations of the 4i, i = 1,2 .... , n, fields, which must leave invariant the free fields equations. Since they are homogeneous transformations, 4n is not required to be massless. We come thus to a linear realization of G on the space V n of the free fields. Since lines are map- ped into lines under linear transformations, the linear realization of G maps the projective space ~ 'n - 1 into ~ ' n - l . However, since we cannot mix the coordinate 4n with the coordinates 4i, i ~ n, (cf. eq. (7)), we must have that the hyperplane in ~ 'n - 1,4n = 0, must be left invariant. This implies that the transformations

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which are allowed in Gin are just those transformations leaving Cn = 0 invariant, and which, restricted to the hyperplane Cn = 0, act as the restriction o f an element in G leaving the hyperplane q~n = 0 invariant. Thus, Gin consists of the (n × n)-matrices

............... 0

a 1 a 2

an- 1 ot

with

I0 . . . . . . . . . . . . . . . . 0

E G .

This situation is what always happens in a very general setting as a geometric consequence of spontaneous breakdown of symmetry. In the cases of physical inter- est we note that this procedure naturally leads to group contraction phenomena due to the minimality of the dimensions of the occurring representations and the compactness of the symmetry groups (cf. ref. [2]).

Let us observe that ~ 'n - 1 can be regarded as the affine space V n - 1 together with an hyperplane at the infinity. From the above considerations then it follows that the spontaneous breakdown symmetry condition acts as to isolate in ~ 'n - 1 the hyperplane at the infinity 4~ n = 0; and this is equivalent to allow inhomogeneous transformations as in eq. (8) for the ¢i fields, i 4: n. It is however well known that such transformations must be understood as the limit for f(x) ~ 1 of the trans- formations

~i(x) ~ ~)i(x) + f(x)ci, (9)

with a square integrable function f(x) (cf. ref. [2])• Then, space integrations are insensitive to locally in-

finitesimal terms of order of 1/V with the volume V ~ . In other words, since the limit f ( x ) ~ 1 acts as an infrared cut-off, infrared effects from the Goldstone fields q~i, i v ~ n, do not contribute to the generators of the transformations in G in . This infra- red effects, which give a finite contribution when in- tegrated over the whole system, are thus the origin of the difference between the dynamical and the observa- tional symmetry groups G and G in, respectively.

In conclusion, by using projective geometry argu- ments, we have shown that in spontaneously broken symmetry theories, the symmetry group Gin that ap- pears in observations is a group contraction of the dy- namical invariance group G. Observable ordered states and low energy theorems are manifestations of the phenomenological symmetry group G in [2]. The dy- namical rearrangement of symmetry which leads to a group contraction occurs in the linearization procedure by which one passes from non-linear Heisenberg equa- tions to the linear free fields equations.

Reference

[1] H. Umezawa, Nuovo Cimento 40 (1965) 450; L. Leplae, R.N. Sen and H. Umezawa, Nuovo Cimento 49 (1967) 1.

[2] C. De Concini and G. Vitiello, Nucl. Phys. Bl16 (1976) 141. [3] I.E. Segal, Duke Math. J. 18 (1951) 221;

E• In/Jn/J and E.P. Wigner, Proc. Nat. Acad. Sci. US 39 (1953) 510.

[4] H. Matsumoto, N.J. Papastamatiou, H. Umezawa and G. Vitiello, Nucl. Phys. B97 (1975) 61.

[5] S. Coleman, Phys. Letters 19 (1965) 144; E. Fabri and L.E. Picasso, Phys. Rev. Letters 16 (1966) 408.

[6] J. Goldstone, Nuovo Cimento 19 (1961) 154; J. Goldstone, A. Salam and S. Weinberg, Phys. Rev. 127 (1962) 965.

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