In all studies of quantum field theory it is assumed that there exists a dis- tinct state (the vacuum state) which is invariant under all translations and.
IL NUOVO CIMENTO
VOL. XXIX, N. 1
1o Luglio 1963
The Vacuum State in Quantum FieId Theory (*). H. J. BORCHERS (**), R. I-IAAG and B. SCHROEIr (%*) Department o] Physics, University o] Illinois - Urbana, Ill. (ricevuto il 3 Gennaio 1963)
- - We wish to show that in a local quantum field theory which describes zero-mass particles the existence of a vacuum state is neither derivable from nor contradicted by the field equations and commutation relations which define the theory. It is an independent postulate which can be added for convenience or left off without changing the essential physical content of the theory. Summary.
1.
-
Introdu~lon.
I n all studies of q u a n t u m field t h e o r y it is assumed t h a t there exists a distinct state (the v a c u u m state) which is invariant under all translations and Lorentz transformations. One might be inclined to question whether this is reasonable assumption if the theory describes, amon~ other things, particles of zero rest mass. F r o m the physical point of view, one might argue t h a t in order to distinguish the v a c u u m from a state with one or several zero-mass particles with v e r y low m o m e n t a one would need a Geiger counter which is m o v e d with almost the velocity of light. Therefore a sharp distinction between the v a c u u m and the neighboring low lying states seems experimentally impossible. Mathematically one finds already in the simple case of a free KleinGordon field a p h e n o m e n o n t h a t corresponds in some way to the above r e m a r k : I f the mass is different from zero, then the Hilbert space of physical states is uniquely characterized b y the c o m m u t a t i o n relations and equations of motion of the field operators and b y the requirement t h a t no negative energies shall occur. This uniqueness statement fails, however, if the mass is zero. I n t h a t case one can find m a n y inequivalent solutions; i.e. the t h e o r y is not deter(') This research was supported in part by the National Science Foundation. ('*) Present address: Physics Department, New York University, New York. ('.') Present address: Institut fiir theoretische Physik der Univerit~t, Hamburg.
THE
VACUUM STATE IN Q U A N T U M F I E L D
THEORY
149
mined b y the c o m m u t a t i o n relations, equations of motion and the requirde positivity of the energy (Section 2). One possible solution is, of course, again Fock space. This is the only one which has a v a c u u m state. The others do not have a n y translationally invariant vectors. I t turns out, however, t h a t this a m b i g u i t y is not v e r y serious. While the various solutions differ in their global behavior, t h e y are essentially equivalent in their predictions for local experiments. I n Section 3 we shall s t u d y this p h e n o m e n o n in a more general context. The discussion there seems to indicate t h a t the assumption of a v a c u u m state is always permissible: if a local field theory does not have a v a c u u m state then it is possible to construct a t h e o r y with v a c u u m state from it. This construction is unique and its physical meaning is simple. Consider as an example the manifold of all states with charge one in electrodynamics. This space of states with charge one is transformed into itself b y the algebra of observables and it clearly contains no normalizable state which is invariant under translations. Hence it gives us an example of a local field t h e o r y without v a c u u m . B u t if we pick out an a r b i t r a r y state vector T and apply to it a translation b y a sufficiently large distance a, then we get a state which is practically equivalent to the v a c u u m as far as measurements of local quantities are concerned. The expectation value of a n y observable which can be measured within the walls of the laboratory, taken in the state T a = exp [ - - i P a ] T is almost equal to the v a c u u m expectation value of the observable if a is large enough. I n t h a t w a y all physical statements pertaining to states of charge zero can be obtained if we know the physics of the states with charge one. The result of our discussion in Section 3 can then be summarized b y saying t h a t the example just mentioned seems to represent the typical situation of a local field t h e o r y without v a c u u m .
2. - Example: system of noninteracting bosons.
We start with the c o m m u t a t i o n relations for the creation and destruction operators (1)
[a(k), at(k')] = 5 ( k -
k') ;
[a(k), a(k')] = [at(k), a+(k')] = 0
and the Hamiltonian
(2)
H -=j~ a*(k)a(k) dak,
where o~ is a function of k.
The linear m o m e n t u m is ~.dven b y
(3)
P =fka+(k) a(k) d~k .
150
It. J . B O R C H E R S , R. HAAG a n d
B. SCHROER
I f we p u t
(~)
~ = v/~
+ m2,
the eqs. (1) and (2) are equivalent to the c o m m u t a t i o n relations a n d field e q u a tion of a Klein-Gordon field. Specifically, one defines the field operators in the usual w a y b y
(5)
A ( x ) = (2u)- t f (a(k) exp [ i ( k x -- wt)] -~ at(k) exp [-- i ( k x -- cot)]) dSk 2-~
Then eqs. (1)-(4) are replaced b y
(6) (7)
([] - t o g a
= 0,
[ A ( x ) , A(y)] ---- i d ( x -
y),
a n d the r e q u i r e m e n t t h a t a space-time translation b y the a-vector a is represented in the t t i l b e r t space b y a u n i t a r y operator U(a) which t r a n s f o r m s the field according to (8)
U ( a ) A ( x ) U-l(a) : A ( x - ~ a) .
I f we t a k e the f o r m (2) for the H a m i l t o n i a n at its face value, then we conclude t h a t H is a positive operator; no negative energies can appear. This is, however, a s o m e w h a t t r i c k y point since (2) can only be regarded as a heuristic definition of H which allows a certain a m o u n t of freedom in its rigorous interp r e t a t i o n (*).
(') Various investigations of the uniqueness of the solution to the scheme (1). (2), (3) exist. An incomplete but representative list is given in refs. (1-4). In the work of ARAKI (which refers to more complicated Hamiltonians) two additional conditions were used, namely the positivity of the energy and the existence of a normalizable vacuum state (discrete eigenstate of H). SHAL~ and SEGAL abandoned the positivity condition but kept the assumption of a normalizable invariant state. Here we shall do the converse, i.e., keep only the positivity condition. It was pointed out to us by B. ZUMINO that the essential content of our first theorem is contained already in Friedrich's book (ref. (i)). Since the terminology and proof technique is somewhat different there we have kept this theorem nevertheless in the present paper. (1) K. 0. FRIEDRICHS: Mathematical Aspects o/ the Quantum Theory o] Fields (New York, 1953). (2) H. ARAKI: Journ. Math. Phys., l, 492 (1960). (3) D. SHALE: Thesis University of Chicago, 1961. (4) I. E. SEGAL: The characterization o] the physical vacuum, to be published.
THE
VACUUM STATE IN QUANTUM FIELD THEORY
151
W e w a n t to exhibit the difference between the cases m :/: 0 and m = 0. Actually our a r g u m e n t does not at all rely on the relativistic invariance of the theol T. So we could just as well t a k e for ~o a n o t h e r n o n n e g a t i v e function of k t h a n t h a t given in eq. (4). The r e l e v a n t distinction for our purpose is w h e t h e r to a t t a i n s the value zero or whether it is s e p a r a t e d f r o m zero b y a finite gap. F o r simplicity of lagnuage we shall, however, stick to eq. (4). The s t a n d a r d r e p r e s e n t a t i o n (Fock representation) of the scheme defined b y eqs. (1), (2), (3) is obtained if there exists a (normalizable) v e c t o r in H i l b e r t space, say To, which is annihilated b y all the a(]e): (9)
a(k)~lo=O
for all k ;
(To, To) = 1 .
Applying the creation operators at(k) r e p e a t e d l y to this vector To one generates the H i l b e r t space. The scalar product between a n y two vectors can be worked out b y m e a n s of the c o m m u t a t i o n relations (1) a n d eq. (9). I t also follows f r o m (3) t h a t To has linear m o m e n t u m zero, i.e., it is i n v a r i a n t under translations in space. This latter p r o p e r t y we shall t a k e always as the definition of the v a c u u m state. The question now is whether the existence of a vector To satisfying (9) can already be derived f r o m (1) and (2) (and the positivity of H ) or whether there are inequivalent alternatives to the F o c k representation. I t is easy to see t h a t for m=/=0 (9) follows indeed from (1) and (2), b u t t h a t this is not the case for m = 0. The positivity of H gives us the restriction for the energy spectrum E o ~ < E < c~ ;
Eo>~0.
I r r e s p e c t i v e of the structure of the s p e c t r u m (whether Eo is a point eigenvalue or the lower end of a continuum) we can choose an arbitrarily small n u m b e r a n d find a normalizable state T which has e x a c t l y zero p r o b a b i l i t y for an energy outside the interval E o < E < Eo + s .
I f one applies a destruction operator
a(/) ~-/a(k) /(k) d3k , on this state then one gets according to (1), (2) a state whose energy is restricted to the interval (10)
152
H.J.
B O R C H E R S , R. HAAG
and
B. SCHROER
where Ik l i s the smallest value of I k t i n the support of the function J. I f m s 0 we need only choose s < m and find t h a t T is annihilated b y all a(J), since then the inequalities (10) become contradictory. Hence T satisfies eq. (9). I f m = 0 we can only conclude t h a t kg is annihilated b y those operators a(]) for which the support of the function ] stays a w a y from the origin by a distance greater t h a n s. Although we m a y choose e as small as we like we c a n n o t obtain a n y information a b o u t the limit e = 0 b y this method. One has, however, the following t h e o r e m :
Theorem I. - The c o m m u t a t i o n relations and field equations of a free Bose field with zero mass allow an infinite v a r i e t y of inequivalent irreducible solutions, all with nonnegative energy. E x a m p l e s of solutions for which the H a m i l tonian (2) and the h n e a r m o m e n t u m (3) h a v e purely continuous s p e c t r u m (no normMizable v a c u u m state exists) are obtainable from the F o c k representation in the following way. L e t b+(k), b(k) be a system of creation and destruction operators in F o c k space and ](k) a numerical function which has a singularity at the origin such that (11)
f []]2d3lc= c~
but
f ]k] I]]2dak < c~.
Put
(12)
a(k) = b(k) -- J(k) ;
a+(k) = bt(k) - - / * ( k ) .
Two such solutions are unitarily equivalent if the difference between the two functions J is square integrable. L e t us add one side r e m a r k a b o u t Lorentz invariance. The defining equations of the t h e o r y are manifestly Lorentz i n v a r i a n t in the form (5), (6), (7). B u t in none of the solutions given b y (5), (12) with / satisfying (11) can we find a u n i t a r y operator U(A) representing the Lorentz t r a n s f o r m a t i o n A such that U ( A ) A ( x ) U-I(A) = A(Ax).
W e have here, therefore, a (rather trivial) e x a m p l e of = Eo(Q)
exists, is i n d e p e n d e n t of T a n d i n d e p e n d e n t of the choice of r e p r e s e n t a t i o n , i.e., it is t h e same for ] = 0 a n d for a r b i t r a r y ] satisfying ( l l ) . I n eq. (13) we h a v e a b b r e v i a t e d (14)
T x = exp[--iP.x]T.
If we t a k e ] = 0 in (8), we come b a c k to the usual F o c k r e p r e s e n t a t i o n of a(k), at(k). T h e n Eo(Q) is t h e v a c u u m e x p e c t a t i o n v a l u e of Q. Therefore T h e o r e m I I tells us t h a t in a n y r e p r e s e n t a t i o n we can find states w h i c h are p r a c t i c a l l y indistinguishable f r o m the v a c u u m state of the F o e k r e p r e s e n t a t i o n as long as we restrict ourselves to quasiloeal onbservables. The inequivalence of the different r e p r e s e n t a t i o n s can therefore show itself only in global measu r e m e n t s a n d is t h u s of little physical interest. This gives a s o m e w h a t u n e x p e c t e d answer to the q u e s t i o n raised in the i n t r o d u c t i o n : is t h e a s s u m p t i o n of a v a c u u m state reasonable or n o t ? W e see t h a t in the p r e s e n t e x a m p l e the a s s u m p t i o n of a v a c u u m is i n d e p e n d e n t f r o m the e q u a t i o n s which define the t h e o r y . W e can a d o p t it or reject it w i t h o u t g e t t i n g a c o n t r a d i c t i o n and w i t h o u t even c h a n g i n g the r e l e v a n t p h y s ical predictions of the t h e o r y . To p r o v e the t h e o r e m s let us use t h e following n o t a t i o n : I f L is a f u n c t i o n of k a n d K = K ( k l , k2) a f u n c t i o n of two a r g u m e n t s (kernel) t h e n we write (15)
(L1, L2) = f L l ( k ) L 2 ( k ) d3k;
(L, K L ) = r E ( k 1 ) K ( k t , k2)L(k2) d3],'~ d3k2 .
(') Actually the following theorem and similar ones are valid for a much broader class of operators. For instance, the operator
Q = f / ( x I ..... xn)A(xl) .... 4 (xn) ddxl ... ddx,~ , would be called quasilocal according to the definition above if the function ] vanishes exactly outside of some finite space-time region. But theorem II holds already if ] vanishes asymptotically for large x, sufficiently rapidly to make f](x 1..... x,) (14xl ... d~xn finite. The term ((asymptotically local )) has been suggested for such an operator but this is perhaps not a very fortunate piece of semantics.
154
n . J . BORCHERS, R. HAAG and s. SCHROER
L e t ]0) be the F o c k v a c u u m and b ( k ) , b~(k) the o r d i n a r y destruction and creation operators. I n other words, we h a v e (16)
b ( k ) 10) = 0
for all k (*). I t is k n o w n t h a t the states (17)
I h> ---- exp [(h, bt)] [0>
belong to F o c k space provided t h a t h is a square integrable function. scwlar p r o d u c t between two such states is (18)
(h'lh)
The
~- exp [(h'*, h)].
I t is also k n o w n t h a t these states provide a complete basis (nonorthogonal, of course) in F o c k space. B y this we m e a n t h a t a n y vector can be approxim a t e d to an a r b i t r a r y degree of precision b y a linear combination of a finite n u m b e r of states ]h). Therefore, in order to define a u n i t a r y operator in F o e k space it is sufficient to define it o a the vectors ]h) a n d to ensure t h a t it conserves the scalar products and ha, s an inverse. We shall do this for the timet r a n s l a t i o n operators (19)
U(t) = exp [ i H t ] ,
with H given b y (2) a n d (12). To m o t i v a t e the definition of U(t) which will be given below (eq. (26)), we shall use the following formal identities:
(20)
e -~ F ( b , b~)e a -~ F ( b ' , b 't) ,
with b'(k) = e-~b(k) e";
b t ' ( k ) -= e - ~ b t ( k ) e ~ .
I f A is a linear form in b and bt: (21)
A ~ - ( L I , b ) ~ (s
t)
thin
(2~')
e-~b(k) e A = b(k) + L~(k);
e-Abe(k) e ~ ---- bt(k) - - L l ( k ) .
(') For clarity let us emphasize again that the ((physical vacuum i) To, if it exists, would be characterized by the property P ~ o = 0 , where P is given by (3) and (12). The ((Fock vacuum,~ 10> is characterized by (16). Obviously ~Oor ]0> unless ](k)=0.
THE
VACUU.~[ S T A T E I N
QUANTUM FIELD
THEORY
155
If A is a bilinear f o r m A = (b t, K b ) ,
(22) then
e-A(L, b)e A = (L, eKb);
(22')
e-A(L, bt)e ~ = (b*, e-KL) .
Finally, we need the f o r m u l a (23)
exp [(L1, b) § (L2, b*)] = e x p [ l ( L x , L2)] exp[(L2, br
exp [(L,, b)].
This collection of formulas is now used as follows. First one sees t h a t p u t t i n g L 2 ( k ) = - - / ( k ) , L d k ) = ]*(k) in (21), (21') we h a v e the t r a n s f o r m a t i o n f r o m the s y s t e m b, b* to a, a* as defined b y (12). Therefore, according to (20) (24)
U(t) = exp [iHt] = exp [--(]*, b) + (/, br exp [i(b*, o~b)t]. 9exp[(/*, b) - - (/, b~)] (*).
W e work out
U(t) Ih> = U(t) exp[(h, b*)] ]0>, b y using (22) to shift the e x p o n e n t i a l of (b*, cob) until it stands i m m e d i a t e l y in front of [0> where it can be omitted. Thus we get
U(t)]h} = exp [-- (]:~, b) + (/, bt)] 9
(25)
9exp [(]*, exp [-- icot]b) -- (b r exp [icot]])] exp [(b*, exp [icot] h)] ]0>. Decomposing the linear e x p o n e n t i a l according to (23) and shifting the dest r u c t i o n p a r t s to the right we get finally (26)
U(t) i h> =- exp IN] ]](1 - - exp [iwt]) -? exp [icot] h>,
with (26')
N = - - (/*, (1 - - exp[io, t])(/-- h)).
Since for a n y t the f u n c t i o n 1 - - e x p [ i c o t ] has a zero of first order at I k ] = 0, it is seen t h a t all quantities appearing on the r i g h t - h a n d side of (26) are well defined and finite, p r o v i d e d t h a t ~/[ k[](k) is square integrable (which was assumed in T h e o r e m I). I t is i m m e d i a t e l y checked with the help of (18) t h a t U(t) as defined b y (26) leaves the scalar p r o d u c t s u n c h a n g e d and t h a t it (') Here r is, of course, the kernel ~(k~, k2)= [kl [6(kl--k2).
156
H.g.
BORCHERS, R. HAAG
has an inverse, namely U(--t). (27)
and
g.
SCHROER
Therefore it is unitary.
U(t~) U(t2) = U(t~ 4- t~) ;
We also check
U(O) =- 1 .
Therefore we can write (28)
U(t) = exp [iHt]
and definite s a self-adjoint operator H. That this operator coincides with the naive definition (2) on a dense set of states is verified b y differentiation of (26). I n the same way we define the linear m o m e n t u m operators. We only have to replace hot in (26) b y - - i k x to get the u n i t a r y operators representing a space translation (29)
U(x) = exp [-- i P x ] .
Note t h a t the condition of square integrability of the function v / i k ] / ( k ) is also necessary for this purpose. Therefore, even if we take a different energym o m e n t u m relation for a single particle, for instance, if we put (30)
(o = k ~
we can still not use functions ] in (21) which are more singular. If we w a n t e d only to define the t t a m i l t o n i a n it would be sufficient to have v / m / square integrable. B u t if we also want the linear m o m e n t a we must have v~lkt/ square integrable as well. The last p a r t of Theorem I which remains to be p r o v e d is the positivity of H. This is most easily tested b y allowing t to become complex in (26). I n particular, we c~n replace t b y i~ and let the real n u m b e r v tend towards 4- c~. I f the n o r m of U(i~)lh} remains bounded in this limit for all [h} then the spectrum of H contains no negative Values. According to (26) and (28) we find
(31)
]tU(iv)lh>[[~=exp[f[]h]~--L/--h]'(1--exp[--2~v] ]dSk I .
I t ] itself is not square integrable then as v --> 4- c~ the exponent approaches - - c ~ so t h a t (32)
lim
T--~co
IIu(i~)lh>a]2 =
0
for all h.
This shows t h a t the energy spectrum is positive and t h a t there is no discrete eigenstate with zero energy. If / were square integrable, then the vector II>
THE
VACUUM STATE IN Q U A N T U M F I E L D
THEORY
157
would be a discrete eigenstate of H to zero eigenvalue and we would have the standard representation of the system a(k), a+(k). Concerning Theorem II, let us calculate, for instance (33)
co. Hence the right-hand side of (33) has the limit exp[(h*, h)§189 L1)]. Dividing by the norm square of the vector [h> we see then t h a t in the notation of (13) (34)
Eo(Q) : exp [89
for Q -- exp [(L2, a+)+ (L1, a)]
which is indeed independent of the state [h> and of the function ] which characterizes the representation. We note t h a t (35)
Eo(exp [(L2, a +) + (/51, a)]) = 9
158
H.J.
B O R C H E R S , R. tIAAG
and
B. S C H R O E R
Fronl (34) we can deduce the E0-value of any product of factors a, a + b y 7r tionM differentiation with respect to the functions L2 and L~. Thus (34) is sufficient to determine Eo(Q) for a n y quasilocM operator Q. I f Q = F ( a , a ~) t h e n we see h'om (35) t h a t
(36)
3. -
Eo(F(e*, a*)) = 4.0 IF(b, b ~) 10>.
General
case.
We assume t h a t we are given a q u a n t u m theory with the following features: 1) A translation in space-time b y the vector a --~ (a, ~) is represented b y the u n i t a r y operator U(a) in the Hilbert space of physical states. We write (37)
U(a) -- exp [ - - i ( P a - - H v ) ] ,
~nd call P the operator of linear m o m e n t u m , H the energy operator. 2) There exists an algebra ~ of
= F~(x).
(*) It is ~ well known theorem that for an angebra of bounded operators tile we-~k closm'e ~nd the strong closure are identical.
160
H. J. BORCHER8,
R. HAAG
and
B. SCHROER
with
F.(x)-
(42')
Interchanging T and q~, replacing Q by Q* ~nd taking the complex conjugate we find
--+ F~(x) and hence
/%(x) -- Fo(x) -+ 0.
(43)
This fact, t h a t the right-hand side of (42') becomes independent of the state T in the limit of large Ix I, is the content of Theorem III. In particular, it implies also (40), since F~,(x + a) = F~, (x)
with
~ ---- U - I ( a ) T .
Equation (40), unfortunately, is not quite strong enough to imply t h a t F(x) approaches a constant as l xl--~ oo. The alternative possibility is t h a t F(x) oscillates with a period increasing to infinity as [x[--> oo. This possibility is very pathological from the physical point of view and we want to exclude it by the Assumption:
lim Fq(x) exists.
Then Theorem I I I can be sharpened to Theorem I V . - For any quasilocM operator Q the sequence Q(x, 3) converges weakly towards a multiple of the identity as Ix[--> oo:
(44)
weak
lim Q(x 3) = ,~(Q).I .
Ixl--+r
We have gained, thereby, a positive linear form over the algebra ~ , which is normalized and translationally invariant: (45)
,~(~QI§247
2(Q*Q)>~0;
4(1)=1;
,~(Q(x))=,~(Q).
Theorem IV is the generalization of Theorem II. We can use the positive linear form ~ to construct a representation of the algebra ~ and of the translation group in a Hilbert space :If' by means of the Gelfand construction. The translational invariance of ~ implies t h a t there is a translationalIy invariant state (vacuum) in 9~'.
TIIE
VACUUM
We shall now p r o v e equation to the other a r b i t r a r y pair of states projection operator on operator Q' such t h a t
STATE
QUANTUM
FIELD
THEORY
](~l
T h e o r e m i I I , or r a t h e r eq. (42). The relation of this s t a t e m e n t s has already been discussed. We pick an q~ a|ld T and a quasilocal operator Q. L e t P be the T. Due to iv) we can find a self-adjoint quasilocal
where e is as small as we want.
= --+ = (q~' T>(TJQ(x)IT>
which is eq. (42). There are two respects in which the present s t u d y is incomplete. First, we feel t h a t the a s s u m p t i o n preceding T h e o r e m I V m u s t be derivable f r o m the usual postulates of q u a n t u m field theory. Secondly, one would like to show t h a t no physical i n f o r m a t i o n is lost if one replaces thc originally given Hilbert space J/f, which had no v a c u u m state in it b y the space 5/f' which is constructed with the help of the linear f o r m 2 and which has a v a c u u m state. T h a t this m u s t be the case is strongly suggested b y the intuitive discussion and, of course, b y the consideration of special examples like t h a t in Section 2. B u t we h a v e not y e t f o u n d a general proof of this feature.
We would like to t h a n k H. ARAKI for some helpful suggestions concerning the proofs of the theorems in Section 2. 11 ~ II N u o v o Cimento.
162
H.J.
Note added
B O R C H E R S ~ R. H A A G
and
B. S C H R O E R
in prof.
D. KASTL]~R has observed that the unsatisfactory features of our discussion in Section 3 can be overcome. Instead of making the assumption which precedes theorem IV, one can prove the following: There is at least one sequence of points xn with ]x~ I--> ~ for n-+ c~ such that lira /~q(xn) exists for all operators Q in the algebra 2. n---~r
From this and theorem I I I it follows that one can construct at least one representation which has a translationaUy invariant state (vacuum). There remains therefore only the question of the uniqueness of the vacuum. Concerning the (( physical equivalence ~>of two representations which are not equivalent b y a u n i t a r y transformation D . KASTLER found the following criterion. All representations of ~ are physically equivalent if and only if the algebra is simple. These matters will be discussed in some detail in a separate paper.
RIASSUNTO
(*)
Si espone un'analisi della produzione di particelle strane nelle collisioni di protoni di 24.5 GeV/e con protoni (energia totale nel s.c.m.=6.72 GeV). Essa si basa su 50000 fotografie prese con la camera a bolle d'idrogeno da 30 cm del CERN. Si fa il eonfronto con u n esperimento sulle interazioni u--p nella stessa camera a bolle. I bassi impulsi trasversali trovati precedentemente rieevono conferma: solo gli iperoni positivi hanno impulsi trasversali elevati. Si di~nno le sezioni d'urto parziali.
(*) Traduzione a eura della Redazionr
