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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)

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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) 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~<E< c~ ; Eo>~0. 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.
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