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)
Summary. -- 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.
1. - Introdu~lon.
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 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. From the physical point of view, one might argue that in order to distinguish the vacuum from a state with one or several zero-mass particles with very low momenta one would need a Geiger counter which is moved with almost the velocity of light. Therefore a sharp distinction between the vacuum and the neighboring low lying states seems experimentally impos- sible. Mathematically one finds already in the simple case of a free Klein- Gordon field a phenomenon that corresponds in some way to the above remark: If the mass is different from zero, then the Hilbert space of physical states is uniquely characterized by the commutation relations and equations of motion of the field operators and by the requirement that no negative energies shall occur. This uniqueness statement fails, however, if the mass is zero. In that case one can find many inequivalent solutions; i.e. the theory 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 QUANTUM FIELD THEORY 149 mined by the commutation 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 vacuum state. The others do not have any translationally invariant vectors. It turns out, however, that this ambiguity is not very serious. While the various solutions differ in their global behavior, they are essentially equivalent in their predictions for local experiments. In Section 3 we shall study this phenomenon in a more general context. The discussion there seems to indicate that the assumption of a vacuum state is always permissible: if a local field theory does not have a vacuum state then it is possible to construct a theory with vacuum 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 by 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 theory without vacuum. But if we pick out an arbitrary state vector T and apply to it a trans- lation by a sufficiently large distance a, then we get a state which is practically equivalent to the vacuum as far as measurements of local quantities are con- cerned. The expectation value of any observable which can be measured within the walls of the laboratory, taken in the state T a = exp [--iPa]T is almost equal to the vacuum expectation value of the observable if a is large enough. In that way 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 by saying that the example just mentioned seems to represent the typical situation of a local field theory without vacuum.
2. - Example: system of noninteracting bosons.
We start with the commutation 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 momentum is ~.dven by
(3) P =fka+(k) a(k) d~k . 150 It. J. BORCHERS, R. HAAG and B. SCHROER
If we put
(~) ~ = v/~ + m2, the eqs. (1) and (2) are equivalent to the commutation relations and field equa- tion of a Klein-Gordon field. Specifically, one defines the field operators in the usual way by
(5) A(x) = (2u)- tf (a(k) exp [i(kx -- wt)] -~ at(k) exp [-- i(kx -- cot)]) dSk2-~
Then eqs. (1)-(4) are replaced by
(6) ([] - toga = 0,
(7) [A(x), A(y)] ---- i d(x- y), and the requirement that a space-time translation by the a-vector a is repre- sented in the ttilbert space by a unitary operator U(a) which transforms the field according to
(8) U(a)A(x) U-l(a) : A(x-~ a) .
If we take the form (2) for the Hamiltonian at its face value, then we con- clude that H is a positive operator; no negative energies can appear. This is, however, a somewhat tricky point since (2) can only be regarded as a heuristic definition of H which allows a certain amount of freedom in its rigorous inter- pretation (*).
(') 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
We want to exhibit the difference between the cases m :/: 0 and m = 0. Actually our argument does not at all rely on the relativistic invariance of the theol T. So we could just as well take for ~o another nonnegative function of k than that given in eq. (4). The relevant distinction for our purpose is whether to attains the value zero or whether it is separated from zero by a finite gap. For simplicity of lagnuage we shall, however, stick to eq. (4). The standard representation (Fock representation) of the scheme defined by eqs. (1), (2), (3) is obtained if there exists a (normalizable) vector in Hilbert space, say To, which is annihilated by all the a(]e):
(9) a(k)~lo=O for all k; (To, To) =1.
Applying the creation operators at(k) repeatedly to this vector To one gene- rates the Hilbert space. The scalar product between any two vectors can be worked out by means of the commutation relations (1) and eq. (9). It also follows from (3) that To has linear momentum zero, i.e., it is invariant under translations in space. This latter property we shall take always as the defi- nition of the vacuum state. The question now is whether the existence of a vector To satisfying (9) can already be derived from (1) and (2) (and the positivity of H) or whether there are inequivalent alternatives to the Fock representation. It is easy to see that for m=/=0 (9) follows indeed from (1) and (2), but that this is not the case for m = 0. The positivity of H gives us the restriction for the energy spectrum
Eo~
Irrespective of the structure of the spectrum (whether Eo is a point eigenvalue or the lower end of a continuum) we can choose an arbitrarily small number and find a normalizable state T which has exactly zero probability for an energy outside the interval
Eo< E < Eo + s .
If 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 re- stricted to the interval
(10) 152 H.J. BORCHERS, R. HAAG and B. SCHROER where Iklis the smallest value of I ktin the support of the function J. If ms 0 we need only choose s < m and find that T is annihilated by all a(J), since then the inequalities (10) become contradictory. Hence T satisfies eq. (9). If m = 0 we can only conclude that kg is annihilated by those operators a(]) for which the support of the function ] stays away from the origin by a dis- tance greater than s. Although we may choose e as small as we like we cannot obtain any information about the limit e= 0 by this method. One has, however, the following theorem:
Theorem I. - The commutation relations and field equations of a free Bose field with zero mass allow an infinite variety of inequivalent irreducible solu- tions, all with nonnegative energy. Examples of solutions for which the Hamil- tonian (2) and the hnear momentum (3) have purely continuous spectrum (no normMizable vacuum state exists) are obtainable from the Fock repre- sentation in the following way. Let b+(k), b(k) be a system of creation and destruction operators in Fock 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. Let us add one side remark about Lorentz invariance. The defining equa- tions of the theory are manifestly Lorentz invariant in the form (5), (6), (7). But in none of the solutions given by (5), (12) with / satisfying (11) can we find a unitary operator U(A) representing the Lorentz transformation A such that
U(A)A(x) U-I(A) = A(Ax).
We have here, therefore, a (rather trivial) example of <~ breaking of a sym- metry ~), a phenomenon frequencly occttring in systems with infinitely many degrees of freedom: The invariance properties of the basic equations ar no longer found in the Hilbert space of states corresponding to a solution of type {12) unless ] is square integrable. Let us now call an operator <~ quasilocal ~> if it is such a functional of the THE VACUUM STATE IN QUANTUM FIELD THEORY 153 field operators A(x) that only space-time points x in some finite region are involved (*). Then we have
Theorem II. - Let T be any state and Q any <~ quasi-local operator )>; then
(13) lim (T~ [(2]T~> = Eo(Q) exists, is independent of T and independent of the choice of representation, i.e., it is the same for ]= 0 and for arbitrary ] satisfying (ll). In eq. (13) we have abbreviated
(14) T x = exp[--iP.x]T.
If we take ]= 0 in (8), we come back to the usual Fock representation of a(k), at(k). Then Eo(Q) is the vacuum expectation value of Q. Therefore Theorem II tells us that in any representation we can find states which are practically indistinguishable from the vacuum state of the Foek representation as long as we restrict ourselves to quasiloeal onbservables. The inequivalence of the different representations can therefore show itself only in global meas- urements and is thus of little physical interest. This gives a somewhat unexpected answer to the question raised in the introduction: is the assumption of a vacuum state reasonable or not? We see that in the present example the assumption of a vacuum is independent from the equations which define the theory. We can adopt it or reject it without getting a contradiction and without even changing the relevant phys- ical predictions of the theory. To prove the theorems let us use the following notation: If L is a function of k and K=K(kl, k2) a function of two arguments (kernel) then we write
(15) (L1, L2) =fLl(k)L2(k) d3k; (L, KL) =rE(k1)K(kt, 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
Let ]0) be the Fock vacuum and b(k), b~(k) the ordinary destruction and creation operators. In other words, we have
(16) b(k) 10) = 0 for all k (*). It is known that the states
(17) I h> ---- exp [(h, bt)] [0> belong to Fock space provided that h is a square integrable function. The scwlar product between two such states is
(18) (h'lh) ~- exp [(h'*, h)].
It is also known that these states provide a complete basis (nonorthogonal, of course) in Fock space. By this we mean that any vector can be approxi- mated to an arbitrary degree of precision by a linear combination of a finite number of states ]h). Therefore, in order to define a unitary operator in Foek space it is sufficient to define it oa the vectors ]h) and to ensure that it con- serves the scalar products and ha, s an inverse. We shall do this for the time- translation operators
(19) U(t) = exp [iHt] , with H given by (2) and (12). To motivate 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"; bt'(k) -= e-~bt(k)e ~ .
If A is a linear form in b and bt:
(21) A~-(LI, b)~ (s t) thin (2~') e-~b(k) e A = b(k) + L~(k); e-Abe(k) e ~ ---- bt(k) -- Ll(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.~[ STATE IN QUANTUM FIELD THEORY 155
If A is a bilinear form (22) A = (b t, Kb), then
(22') e-A(L, b)e A = (L, eKb); e-A(L, bt)e ~ = (b*, e-KL) .
Finally, we need the formula
(23) exp [(L1, b) § (L2, b*)] = exp[l(Lx, L2)] exp[(L2, br exp [(L,, b)].
This collection of formulas is now used as follows. First one sees that putting L2(k)=--/(k), Ldk)= ]*(k) in (21), (21') we have the transformation from the system b, b* to a, a* as defined by (12). Therefore, according to (20)
(24) U(t) = exp [iHt] = exp [--(]*, b) + (/, br exp [i(b*, o~b)t].
exp[(/*, b) -- (/, b~)] (*). We work out
U(t) Ih> = U(t) exp[(h, b*)] ]0>, by using (22) to shift the exponential of (b*, cob) until it stands immediately in front of [0> where it can be omitted. Thus we get
(25) U(t)]h} = exp [-- (]:~, b) + (/, bt)]
exp [(]*, exp [-- icot]b) -- (b r exp [icot]])] exp [(b*, exp [icot] h)] ]0>.
Decomposing the linear exponential according to (23) and shifting the des- truction parts 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 any t the function 1--exp[icot] has a zero of first order at I k]= 0, it is seen that all quantities appearing on the right-hand side of (26) are well defined and finite, provided that ~/[ k[](k) is square integrable (which was assumed in Theorem I). It is immediately checked with the help of (18) that U(t) as defined by (26) leaves the scalar products unchanged and that it
(') Here r is, of course, the kernel ~(k~, k2)= [kl [6(kl--k2). 156 H.g. BORCHERS, R. HAAG and g. SCHROER has an inverse, namely U(--t). Therefore it is unitary. We also check
(27) U(t~) U(t2) = U(t~ 4- t~) ; 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 by differen- tiation of (26). In the same way we define the linear momentum operators. We only have to replace hot in (26) by --ikx to get the unitary operators representing a space translation
(29) U(x) = exp [-- i Px] .
Note that the condition of square integrability of the function v/ik]/(k) is also necessary for this purpose. Therefore, even if we take a different energy- momentum 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 wanted only to define the ttamiltonian it would be sufficient to have v/m/ square integrable. But if we also want the linear momenta we must have v~lkt/ square integrable as well. The last part of Theorem I which remains to be proved is the positivity of H. This is most easily tested by allowing t to become complex in (26). In particular, we c~n replace t by i~ and let the real number v tend towards 4- c~. If the norm 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 .
It ] itself is not square integrable then as v --> 4- c~ the exponent approaches --c~ so that
(32) lim II u(i~)lh>a]2 = 0 for all h. T--~co
This shows that the energy spectrum is positive and that there is no discrete eigenstate with zero energy. If / were square integrable, then the vector II> THE VACUUM STATE IN QUANTUM FIELD 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) This corresponds to the choice of Q = exp [L2, a +] exp [L1, a] in (13). If Q is quasilocal then the functions L~(k) and Ll(k) can have no worse singularity than C.Ik[ - We see this from (5), since A(x)](x) d'x = (g, a) ~- (h, a+), with g(k) - ?(k, l kl) h(k) - ?(- k, -Ikl) 24k! ' 21kl If / has a finite support in x-space then f is regular in k-space. If L has a [k[ -t singularity the functions ].L and a ]ortiori k.L are still absolutely integrable at the origin in k-space so that by the Riemann- Lebesgue Lemma f[] [exp [- ikx] Ll.~(k) d3k and [ h ]exp [-- ikx]L~.~(k) dak, vanish in the limit Ix[--> 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 that in the notation of (13) (34) Eo(Q) : exp [ for Q -- exp [(L2, a+)+ (L1, a)] which is indeed independent of the state [h> and of the function ] which char- acterizes the representation. We note that (35) Eo(exp [(L2, a +) + (/51, a)]) = <0 ]exp [(L~, b +) -t- (L1, b)] 10> 158 H.J. BORCHERS, R. tIAAG and B. SCHROER Fronl (34) we can deduce the E0-value of any product of factors a, a + by 7r tionM differentiation with respect to the functions L2 and L~. Thus (34) is sufficient to determine Eo(Q) for any quasilocM operator Q. If Q=F(a, a ~) then we see h'om (35) that (36) Eo(F(e*, a*)) = 4.0 IF(b, b ~) 10>. 3. - General case. We assume that we are given a quantum theory with the following features: 1) A translation in space-time by the vector a --~ (a, ~) is represented by the unitary operator U(a) in the Hilbert space of physical states. We write (37) U(a) -- exp [-- i(Pa -- Hv)], ~nd call P the operator of linear momentum, H the energy operator. 2) There exists an algebra ~ of << quasi-locM operators ~> which contains essentially all observables. This second assumption needs some explanation. Intuitively speaking, the term << quasi-locM observable >> shall mean that the quantity can be measured within some finite region of space-time. It does not matter how large this region is as long as it is not infinitely extended. Mathematically there are two alternative ways to arrive at a definition of ~. The first one connects ~ with the field operators of a local field theory. In such a theory one may associate with every region B of space-time an algebra of observables R B as explained in earlier papers (5). Then we may define ~ as the union of all those R 8 for which the region B is finitely extended (compact). In other words, Q E~ means that there exists a compact region B such that Q~R B. The other alternative is to define ~ abstractly by its relevant properties. We shall need the following: i) ~ is an algebra. This means that if Q and Q' belong to ~ and if is ~ complex number then Q*, aQ, Q+Q', Q.Q' also belong to ~. For sim- plicity we shall take ~ to consist only of bounded operators although this restriction can have only very little relevance to our problem and should be regarded merely as a technical device of eliminating pathologies which other- wise would need an extensive discussion. (~) R. HAAG: Colloque I~d. sur les probl~mes math. de la th~orie q~tatttique des champs Lille 1957, ed. by CNRS (Paris, 1959); R. HAAG and B. SCHROEZ: Jourl~. Math. Phys., 3, 248 (1962). TIIE VACUUM STATE IN QUANTUM FIELD THEORY 159 ii) ~ is stable under finite translations: From Q6~ it follows tha~ Q(x) ~ ~ where (38) Q(x) = U(x) Q U-~(x) . iii) For ever pair Q, Q' from ~ we have (39) lira [Q(x, ~), Q'] o. Ixl~-->co iv) The weak closure of ~ tzives the algebra of all bounded operators in the Hilbert space. This statement was meant above by the phrase that shall contain essentially all observables. If ~ is defined concretely in the first mentioned way, then it is immediately clear that all the properties listed above are realized. In the case of super- selection rule ~f has, of course, to be interpreted as a single coherent subspace. We state now the following theorem. Theorem III. - For any quasilocal Q we can find a numerical function Fr (x) such that the difference Q(x)--F~(x) converges weakly towards zero as Ix]-->cr The function Fq(x) has the property (40) ~(x + a) - F~(x) ~ 0 as l* I ~ ~o. Before proving this theorem let us comment on its simple physical meaning. An equivalent formulation would be Theorem III'.- For any two quasiloeal operators Q1 and Q~ and any state T we have (41) In other words, two measurements in far separated regions are statistically independent. It is trivial to derive III' from III. In the opposite direction we have to make use of iv) which tells us that by the application of a quasi- local operator on T we can get arbitrarily close to any vector r (*). Therefore (41) can also be written (42) (*) 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 Interchanging T and q~, replacing Q by Q* ~nd taking the complex conjugate we find <~lQ(x) IT> --+ F~(x) <~, ~> and hence (43) /%(x) -- Fo(x) -+ 0. This fact, that 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 im- plies also (40), since F~,(x + a) = F~, (x) with ~ ---- U-I(a)T. Equation (40), unfortunately, is not quite strong enough to imply that F(x) approaches a constant as l xl--~ oo. The alternative possibility is that 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 III can be sharpened to Theorem IV. - For any quasilocM operator Q the sequence Q(x, 3) con- verges 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 trans- lation group in a Hilbert space :If' by means of the Gelfand construction. The translational invariance of ~ implies that there is a translationalIy inva- riant state (vacuum) in 9~'. TIIE VACUUM STATE IN QUANTUM FIELD THEORY ](~l We shall now prove Theorem iII, or rather eq. (42). The relation of this equation to the other statements has already been discussed. We pick an arbitrary pair of states q~ a|ld T and a quasilocal operator Q. Let P be the projection operator on T. Due to iv) we can find a self-adjoint quasilocal operator Q' such that where e is as small as we want. Then we have where and 5~(x) -+ 0 as x -+ o0. The quantities ~ and ~3 are independent of x since the norm of the operator Q(x) is the same as that of (2 and they can be made arbitrarily small by a suitable choice of (2'. Hence we have We would like to thank H. ARAKI for some helpful suggestions concerning the proofs of the theorems in Section 2. 11 ~ II Nuovo Cimento. 162 H.J. BORCHERS~ R. HAAG and B. SCHROER Note added 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 the- orem 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 III it follows that one can construct at least one repre- sentation which has a translationaUy invariant state (vacuum). There remains there- fore only the question of the uniqueness of the vacuum. Concerning the (( physical equivalence ~> of two representations which are not equi- valent by a unitary 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 un 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