INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

Transcription

1 IC/71/ <. : i INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS A FORMULATION OP QUAWTUt^ FIELD THEORY IN TEEMS OF TBT-IPBRED DISTRIBUTIONS AND A POSSIBLE DEPASTURE FROK TEE PRINCIPLE OF SUPERPOSITION J. Rayski INTERNATIONAL ATOMIC ENERGY AGENCY -UNITED NATIONS EDUCATIONAL. SCIENTIFIC AND CULTURAL ORGANIZATION 1971 MIRAM ARE-TRIESTE

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3 IC/T1/71 International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS A FORMULATION OP QUANTUM FIELD THEORY IN TERMS OF TEMPERED DISTRIBUTIONS AND A POSSIBLE DEPARTURE FROM TEE PRINCIPLE OF SUPERPOSITION J. Rayski ABSTRACT A new type of representations related to complete sets of square integrable functions is introduced. The Green's functions of the theory Decome tempered distributions. Normal ordering of creation and annihilation operators related to such representations yields Hamiltonians which are no longer positive definite. This difficulty is circumvented by introducing suitable supplementary conditions. The principle of superposition of states bcreaks down under such supplementary conditions but a probabilistic interpretation in terms of conditioned probabilities is still possible. The difficulties with an infinite volume are avoided. The cases of massive and massless particles may be treated on an equal footing. The old difficulty with securing the Lorentz condition is avoided without having to introduce a non-definite metric. A large class of non-polynomial (e.g. square root) lagrangians becomes quantizable along these lines. MIRAMARE - TRIESTE July 1971 On leave of absence from the Institute of Physics, Jagellonian University, Cracow, Poland.

4 ACOOWLEDCSlHSfTS The author is grateful to Professor Abdus Salarn, the International Atomic Energy Agency and U5TESCO for hospitality at the International Centre for Theoretical Physics, Trieste. -f

5 I. INTRODUCTION In all previous attempts at formulating a quantum theory of fields it was believed to be obvious and indisputable that - formally - this theory should be exactly the same as ordinary quantum mechanics, the only difference being that in one case one has to do with a finite and in the other case with an infinite manifold of degrees of freedom. The field quantities are regarded as forming a continuum of generalized coordinates V. The theory is assumed to be derivable from a variational a principle with a lagrangian dependent on field quantities and their firstorder derivatives. It is a canonical theory, i.e., there exist momenta TT canonically conjugate to the field quantities. Field quantization consists in replacing the field quantities and their canonically conjugate momenta by operators satisfying the following (equal-time ) commutation relations: constituting (at least in the case of commutators) a natural generalization of the ordinary, canonical commutation relations, well known from the case of systems with a finite number of degrees of freedom. The requirements of special relativity are satisfied ~hj assuming the field quantities to be geometrical objects in Minkowski space and the langrangian density to be a scalar with respect to the Poincare group. The appearance of a 6- distribution to the right-hand side of Eq. (l) guarantees microcausality. In close analogy with the case of quantum mechanics, it was taken for granted that the operators U 1 and ir span a Hilbert space and it wr\s a a regarded as self-evident that all (well normalized)vectors in this space describe possible(quantum) states of the field. The principle of superposition, valid for vectors forming a linear (Hilbert) space, enabled the usual probabilistic interpretation of all possible results of field measurements. In order to obtain a physically meaningful theory, it was considered necessary to impose some restrictions upon the lagrangian: viz,, it should be of such a form as to guarantee a positive definite Hamiltonian. It vras believed that examples of fields do exist which satisfy all the above requirements in both cases, those of free fields and those of interacting fields. Moreover - exaotly as in the case of ordinary quantum mechanics - it was expected that any two constructions (representations) of field operators satisfying the same relations (l) are unitarily equivalent -2-

6 provided one assumes the tfamiltonian to he the same or, at most, differing by an additive oonstant. However, after more than forty years of investigations of quantum field theories, no single example of interacting fields (in a fourdimensional Minkowski rpace ) is known to he - with certainty - satisfactory, i.e., free of infinities arid other ambiguities and, at the same time, nontrivial, viz., not being reducible to the case of free fields. This should be regarded as a warning and a strong indication that, in principle, it must "be impossible to maintain all the above assumptions and to transplant automatically the axioms forming the basis of quantum mechanics to the case of field theory. A sacrifice of at least one of the usual quantumtheoretical axioms - hitherto believed to be unavoidable - seems absolutely necessary. II. REPRESENTATIONS CONNECTED WITH TEMPERED DISTRIBUTIONS An investigation of the mathematical structure of the theory of quantized fields, undertaken by several mathematically oriented physicists like Haag, Wightman, Jaffe and others, has revealed deep ambiguities inherent in the fundamental objects (operator-valued distributions) of quantum field theory and an urgent need of a more precise formulation of these concepts as well as the rules of how to deal with them in practical computations. In our opinion, the discovery of the existence of inequivalent representations is of particular importance because it shows that, even if several representations were mathematically correct (would lead to no inconsistencies), we should still have to make a choice between them because they would yield physically inequivalent theories. In particular, the problem appears of whether one should insist upon Fook type representations, i.e., representations of the numbers of particles (with some detailed dynamical characteristics) obtained by applying the respective creation operators to a vacuum state (the latter being understood as an eigenstate of energy) or whether one should rather look for some other representations inequivalent to Fock representations. A suggestive possibility of constructing representations of quite a different type is the following. First of all, instead of allowing for quite arbitrary decompositions of the field quantities into any complete sets of functions, we may introduce a priori a restriction by assuming the following axiom. 3

7 Physical fields are only those representable as a series n where \p denotes a complete set of square integrable functions in the whole three-dimensional space. Inasmuch as any square integrable function expands uniquely into a hermitian series n converging unconditionally in square mean, we may limit our considerations to the hermitian series, i.e., expansions in terms of eigenfunotions of a harmonio oscillator: where y(7, t) = \ a (t )h (r) (4) n n with f' n^ denoting the derivative of the order n while the index n denotes an abbreviation for Now, the quantization consists in replacing the enumerable set of coefficients ^(t) and their conjugate momenta p (t) by operators satisfying the relations ] - 16^. (7) (The indices a, b enumerating the field quantities are dropped for the sake of simplicity of notation.) In this way, the separability of the Hilbert spaoe is ensured. Moreover, by restricting the class of functional representations of the fields, the o-symbol appearing to the right-hand

8 side of Eq. (l) beoomes a tempered distribution. Generally, any object encountered in quantum field theory becomes an operator-valued tempered distribution, in particular the Green's functions appearing in the theory are tempered distributions. Now we may proceed to formulate a representation of the numbers IT of particles with detailed characteristics n, i.e., localized in the domains where the hermitian functions are essentially different from zero and, in consequence of the uncertainty relations, whose squared momenta and kinetic energies are not sharply defined. This has the following advantages! one neither needs to use a box of periodioity, nor to introduce any cut-off in the x-space. The operators constructed exclusively of the respective creation and annihilation operators a, a are meaningful without the necessity of introducing any rapidly decreasing test functions or any procedure of averaging. All the famous difficulties connected with the infinite volume disappear I Of oourse, any other representation that may be proved to be unitarily equivalent to the former is equally well acceptable as a physically admissible representation. It is not difficult to show that two representations in terms of hermitian functions translated by (infinitesimal) constants x^-*x JX + with respect to each, other are connected by a unitary transformation, i.e. are equivalent. Thus, the formalism is translationally invariant. In order to show that it is also Lorentz covariant, it is sufficient to replace the family of coordinate hyperplanes t = constant by an arbitrary one-parametric family of hyperplanes with a time-like normal. The above representations are essentially different from Fock representations. Ifo eigenstate of a definite number of particles, not even a "vacuum-state" H» 0, is an eigenstate of energy, not even in the case of free fields I It is so because the detailed characteristics n of the particles is not that of their (sharp values of) momenta and kinetic energies. Consequently, the construction of these representations does not start from the assumption of "existence of a (lowest and discrete) value within the spectrum of energy of the field. The "vacuum *) J. Mikusinski, Bull. Acad. Polon. Sci. 16., 9, 727 (1968). -5-

9 state" in this representation (if = 0) is not an eigenstate of energy. An immediate advantage of such a formulation is that one can apply it also to the case of massless particles where there is no gap "between the lowest energy level and the rest of the energy spectrum. I l l. A BHEAK-DOWN OF THE SUPERPOSITION PRINCIPLE A serious difficulty appears which, most prohably, constituted the main reason why none of the previous authors has put forward the possibility of introducing such or similar representations. Namely, if one applies Wick's normal ordering of creation and annihilation operators a +, a (which means subtraction of an infinite constant from the tfamiltonian) one is subtracting more than in the case of ordering the usual operators a,, a, (creation and annihilation operators of particles with sharp values of their momenta). Consequently, one obtains a new Hamiltonian which is no longer positive definite, or even bounded from below. This could be regarded as a decisive obstacle preventing the use of such representations. But is it really so? Is it absolutely necessary to have a Hamiltonian whose spectrum is positive definite? The main aim of the present paper is to point out a possibility of formulating a satisfactory theory with a non-definite energy spectrum. The point is that the condition for a positive definiteness of energy does not need to be regarded as a condition upon the operator (Hamiltonian) itself, but rather as a condition upon the state vector. First of all, it is to be remembered that the requirement of a positive definite energy is Lorentz covariant if and only if the fourmomentum is not space-like. Thus, one has to secure first M 2 - E 2 - P* 2 > 0 (8) and then Inasmuch as both M 2 E > 0 (9) and E are constants of the motion, the above two conditions may be satisfied (even in the case of operators whose spectra are indefinite) as initial conditions upon the state vectors. However, since the well normalized states cannot be, in general, eigenbtates of 2 l\ and E and, in particular, the basis vectors of the above representation -6-

10 N_S are not, the above conditions must be understood to hold in a weak sense, as holding true for the expectation values. Thus, we introduce the following axiom. oonditions: Physical states $ ^ are only those which satisfy the following V 0 and < <j? E 1 $>/ > 0. (10) As mentioned above, these conditions are automatically preserved in the course of time if they have been satisfied at an initial instant, say t => 0, Consequently, they do not constitute any encroachment upon the dynamics. On the other hand, they mean severe encroachments upon the theory of measurements of observables. This is so because the manifold of physical states restricted in the above way does not constitute a linear space any longer. Indeed, a linear combination of physical states does not need to be a physical state again. In particular, forming a hermitian representation j IT \ in the oase of a real scalar free field, one easily finds that all vectors of the basis satisfy the above requirements (10), viz. where <^N 0,N,-«- :Hi N 0,N lt -"> - JZ K >n > n (ll) Ci = \in + \n + 5/d )*- ) \±c-) with m denoting the mass constant appearing in the Klein Gordon equation and K. is the characteristic frequency appearing in Eq. (5)» However, the usual formula (ll) only holds true as an expectation value, while N 0,N,,»» are not eigenvectors of H. The energy of the field is not a diagonal matrix in this representation and just the presence of non-diagonal matrix elements has the effect that for some linear combinations of the basis vectors the expectation values of energy are negative. ) Even in the case of a free field, energy consists of two terms IP ' + H f where the quasi-interaction part H* assumes (in the onedimensional case) the form H'

11 Consequently, some restrictions upon possible results of measurements are needed and have to be introduced a priori. Such restrictions are physically interpretable in the following way: i) No interaction with a measuring apparatus can violate the measured system to such an extent as to force it into a state of motion where it would move (as an isolated system after the measurement has been completed) with a velocity greater than that of light, ii) No measurement device is able to take off from *) the measured system more energy than it possessed 'beforehand. Thus, there is some information available prior to any measurement and, consequently, we have to deal with conditioned probabilities. Accordingly, the probabilistic interpretation must be modified. The probabilities of transitions (caused by the act of measurement) from physical to unphysical states are a priori zero whereas the squared absolute values of the scalar products \< ]A ^ mean relative probabilities for obtaining in a measurement of A a physical state I A ^ if the (physical) state just before the measurement in question was [ $\ Thus the probabilities for obtaining in a measurement of A a state A^^> are n according to whether A / is a physical or an unphysical state, with where the summation is only extended over physical states. In this way it is seen that the principle of superposition is a sufficient but not a necessary condition for a probabilistic interpretation. IV. SOME APPLICATIONS Hy relaxing the ordinary quantum-mechanical axiom called the principle of superposition of states and applying, instead, the concept of conditioned probabilities, one acquires a possibility of a much wider choice of both field-theoretical models (that are quantizable) and possible representations *) A full justification of such statements is provided by a new interpretation of measurements in quantum theories presented by the author in a separate publication ('ICTP, Trieste, preprint IC/7l/44 (l97l)).

12 to be introduced. First of all, there appears a possibility of using representations in terms of the numbers of particles N whose detailed dynamical characteristics are connected with square integrable functions, in particular with eigenfunctions h of a harmonic oscillator. Besides the advantages consisting in guaranteeing a separable Hilbert space and avoiding the usual difficulties with the infinite volume, there appears a possibility of treating, in this way, on equal footing the cases of particles with or without a rest mass. Another application is offered as a new possibility of dealing with the Lorentz condition. Instead of assuming the state to be an eigenstate of the operator <L A^ to the eigenvalue zero which cannot be satisfied by any normalizable state vector unless one uses a desperate way out consisting in introducing an indefinite metric in Hilbert space (Bleuler and Gupta), we may satisfy it in the weak form '/* 1 / " Of course, an arbitrary superposition of states satisfying this condition does not need to satisfy it also, but it is again the same situation as that described in Section III, and may be mastered by the same device: defining a sub-class of physical states satisfying Eq. (14) and using conditioned probabilities. But the most important application seems to be offered by the fact that we are able now to consider such models of fields which, hitherto, seemed to be non-quantisable, e.g. a field of Born-Infeld typo with a Lagrangian of a square root form, e.g. (15) where cz 0 is any conventional (polynomial) lagrangian considered in the traditional field theory. The Hamiltonian corresponding to such A lagrangian ceases to be positive definite if Wick's ordering is introduced (it is not even hermitian in the whole domain of its arguments) but, again, we may restrict the manifold of states to physical ones, satisfying the conditions (10), and formulate a strongly non-linear, non-polynomial but physically meaningful quantum field theory«it will be discussed in more detail in a separate publication. *' This is connected with the fact that, according to (ll) and (12) there is a gap between the energy expectation values between the states NQ = 0 and NQ = 1 even if m = 0-9-