Conceptual Problems in Quantum Field Theory

Brazilian Journal of Physics, vol. 34, no. 4A, December, 2004 1343

Walter F. Wreszinski Instituto de F´ısica, Universidade de Sao˜ Paulo, Caixa Postal 66318, 05315-970 Sao˜ Paulo, Brazil

Received on 26 November, 2003

We review some conceptual problems in quantum ﬁeld theory, with an emphasis on exact results and open problems.

We review some conceptual problems in quantum ﬁ- 2. Nonperturbative results and problems: Triviality of the eld theory, with an emphasis on exact results and open S matrix and nonperturbative construction of S(g) . Open problems. The review is organized as follows: 1. The problems. Stuckelberg¨ - Bogoliubov - Shirkov - Epstein - Glaser S(g) operator. A brief review of perturbative results, including the adiabatic limit. Open problems: the infrared problem 1 Stuckelberg-Bogoliubov-Shirkov-¨ in QCD and its relation to gauge invariance. The problem of convergence and asymptotic character of the S-matrix Epstein-Glaser S(g) operator. series: the problem of large coupling. Problems with the se- In perturbation theory S(g) is the generating functional for ries for (large and) small coupling: deﬁnition and triviality. the time-ordered products of Wick polynomials,

Z X∞ 1 S(g) = 1 + d4x . . . d4x T (x , . . . , x ) · g (x ) · (g ) . . . g (x ) , n! 1 n n 1 n 1 2 n n=1 where g ∈ D(R4) is the “adiabatic switching”. d

The n-point operator-valued distributions Tn are the basic What are the advantages of working with S(g)? objects of the theory. They are constructed inductively from Less is required than the Wightman axioms; for g ∈ D, T1 , through causality, unitarity and Lorentz-covariance: we may work in Fock space, the infrared problem is decou- a) S(g)−1 = S(g)∗ (unitary) pled and arises only in the adiabatic limit g → 1. However: the main properties of general quantum field theory are pre- b) S (g + g ) = S (g ) S (g ) if 1 2 1 2 served, in contrast to “effective” field theories. a. supp g > supp g (this ordering refers to the time- Let now T1(x) be the interaction Lagrangian in first or- 1 2 u coordinate) der (e.g. T1(x) = ie : ψ(x)γ ψ(x): Aµ(x) for QED: all fields free fields on F ). It turns out that by causality b. supp g1 ∼ supp g2 (causality) Where “∼” means spaceline; Tn (x1, . . . , xn) = T (T1(x1) ...Ti(xn)) (1) c. There exists a unitary representation U(a, ∧) of the ↑ Poincare´ group P+ on Fock space F (the free field away from the total diagonal representation) such that (Poincare´ covariance) ∆n (x1, . . . , xn) = {(x1, . . . , xn)(x1 = x2 = ... = xn)} . U(a, ∧)S(g)U(a, ∧)−1 = S({a, ∧}g) . (2) The extension of the T (x1, . . . , xn) to the total diagonal is Above, supp g means the (space-time) support of g , assu- not unique and is fixed by Poincare´ covariance, unitarity and med to be an infinitely differentiable function of compact (re) normalizability: the scaling degree of each local term, 4 support, a class denoted by D(R ) – and obtained by normalizing Tn, is ≤ 4. ¡ ¢ This scaling degree – also called Steinmann degree – ({a, ∧}g)(x) ≡ g ∧−1(x − a) . is defined in the process of distribution splitting, a basic 1344 Walter F. Wreszinski

0 method used in causal perturbation theory ([1-3]), which is desired expansion A µ(x) = Aµ(x) + λ∂µ u(x) + 0(λ2) , auxiliary in the determination of Tn . A convenient sum- which is suggested by classical field theory, except that now mary may be found in [4]. This degree is related to Wein- u(x) is, naturally, assumed to be a free quantum field with berg’s power counting [11]. Together with an additional re- the same mass as Aµ , we conclude that quirement – perturbative quantum gauge invariance – the µ µ Tn are determined also in nonabelian gauge theories, but the [Q, A (x)] = i∂ u(x) (4) “gauge group” – with antissymetric structure constants – is derived from the above axioms [3]. It is to be emphasized which determines Q up to a c-number, due to the irreduci- that the requirement of perturbative quantum gauge invari- bility of the Fock representation. It turns out [3] that u(x) is ance is purely quantum, without any reference to classical a Fermionic ghost field, in order that Q have the important physics. It involves a gauge charge Q : property of being nilpotent:

0 2 A µ(x) = e−iλQ Aµ(x)eiλQ (3) Q = 0 (5) wich may be expanded by means of the Lie series Aµ(x) − As remarked, the method also works (with simplifications) iλ [Q, Aµ(x)] + 0(λ2), where Aµ(x) is a free (massive or for massive gauge theories [6]. Quantize the free vector fi- µ massless) vector field in the theory. Comparing with the elds (Aa )a=1,...,M by

¡ 2¢ a µ ν µν µ∗ µ ¤ + ma Aµ = 0; [Aa (x),Ab (y)] = ig δa,b∆m(x − y),Aa = Aa , (6)

where ∆m is the Pauli-Jordan distribution of mass m, where the Ten (x1, . . . , xn) are the n-point distributions which corresponds to the Feynman gauge. The represen- corresponding, however, to S(g)−1 instead of S(g) tation of this *-algebra requires an indefinite inner product ([3], pg. 127). By means of (5) one easily finds [3] that space. It is thus natural to work in a Krein Fock space F, ³ \ ´ denoting the usual scalar product in F by (·, ·) and A+ the D = ran Q ⊕ ker Q ker Q+ ⊕ ran Q+ . adjoint of A with respect to (·, ·) . Let J be the Krein operator: J 2 = 1; J + = J . Then the indefinite inner product In addition it is assumed that

h·, ·i is deﬁned by T Jker Q+ ker Q = 1 (positivity assumption). (11) hψ, φi ≡ (ψ, Jφ) ψ, φ ∈ F (7) Then the h·, ·i – product is positive deﬁnite on \ and * denotes the adjoint with respect to h·, ·i: + Hphys = ker Q ker Q = ker Q/Ran Q , (12)

0∗ = J0+J; h0ψ, φi = hψ, 0∗Φi . (8) which is interpreted as the physical subspace of F: it does not contain the unphysical degrees of freedom [3]. From ∗ For the ghost ﬁelds, ua(x) = ua(x) and for the an- this and (10) the usual unitarity a) on Hphys follows ([3], + ∗ tighost ﬁelds uea(x)(6= ua (x)!) , uea(x) = −uea(x) ([3], pg. 128). This structure, with the Hermiticity condition (9) pg. 126). Convergence of the integral for Q (which results is most clearly expounded in [6], (which we followed), in from (4)), which it is also shown that for massive theories the condition Z X of perturbative quantum gauge invariance may be replaced 3 ν ~ by the physical consistency of the S matrix Q = d x ∂ν Aa(x)∂0ua(x), a [Q, S]/ker Q = 0 , (13) (massless case) follows by a method due to Requardt [19]. ν∗ ν ∗ where S– the true S matrix – is the adiabatic limit of S(g) Notice that by Aa = Aa and ua(x) = ua(x), Q above is an unbounded ∗–symmetrical nilpotent operator on F: Sψ = lim S(g∈)ψ (14) ∈→0 ∗ Q = Q (on a dense invariant domain D) (9) 4 for all ψ ∈ D , where g0 ∈ D(R ), and

This structure implies perturbative pseudo unitarity in g∈(x) ≡ g0(∈ x) , (15) the Krein-Fock Hilbert space, deﬁned as being the following property where g ≡ g0(0) > 0 is the (physical) coupling constant. For massive theories the existence of the adiabatic limit was ∗ Tn (x1, . . . , xn) = Ten (x1, . . . , xn) (10) proved by Epstein and Glaser [7]. Brazilian Journal of Physics, vol. 34, no. 4A, December, 2004 1345

From theories with massless particles, a more elaborate It seems that in QCD there is no cancellation, but exponen- condition is required ([3], pg. 99), which is equivalent to tiation holds. Thus D = −∞,F0 vanishes. Since there the Ward-Takahashi identity for QED ([3], pg. 100) and is no cancellation, physical probabilities vanish as well – as which, for massless SU(N) - Yang-Mills theories, implies far as perturbation theory is, at least, asymptotic – and one the Slavnov Taylor identities ([20], [16]). speaks of perturbative infrared slavery. This is an unproved The differences regarding the usual BRST charge are conjecture for qcd, at least for initial states which are not ([3], pg. 240): a) the BRST variation of the total Lagran- colour singlets; see [12]. In refs. [13] and [14] the authors gian is zero, but the gauge variation of the interaction T1, start with the remark that individual Feynman graphs, are only, is not zero but a divergence; b) the BRST transfor- not gauge invariant, but that appropriate sums (“invariance mation operates on the interacting fields whereas the gauge classes”), are: for instance, for second-order Compton scat- variation transforms free fields. See, however, recent more tering, the sum of the two diagrams where the external pho- general formulations [5]. ton lines are interchanged. What are the advantages of causal perturbation theory? They generate these invariance classes in QED by loo- First and foremost is the treatment of gravity [3]: in the tra- king at Green’s functions of a “dressed” electron field χ, ditional approach, with ultraviolet regularization, it is un- R µ clear, due to non-renormalizability, how to obtain cutoff- χ = e f Aµτ ψ , (19) independent finite results. In the causal theory one has only 0 0 to fix finite (normalization) constants obtained in the pro- where τ is a translation operator τp0 → p − k acting on µ cess of distribution splitting. Of course, the number of finite momentum space, f is a chosen function and ψ is the constants grows with each order, and it is unclear how to fix usual electron field, thus given by the nonlocal transforma- them; but this (open) problem is of a less crucial nature. Se- tion (the inverse of (19)): condly there is the conceptual clarity: for instance, the exis- Z 0 4 0 0 0 0 tence of a scalar field (“Higgs field”) is also derived from ψ(p ) = d q S (p , q ,Aµ) χ(q ) the “axioms”, without assuming spontaneous breakdown of symmetry. Since gauge symmetry is a local symmetry, it with cannot be broken (Elitzur’s theorem, see [8]), and it seems · Z ¸ very artificial to introduce an extra global symmetry which S (p0, q0,A ) = exp −e d4kf (q0, k)Aµ(k).τ is, then, broken. Finally, for some superrenormalizable the- µ µ ories, a mechanism of dynamical mass generation may take place, e.g., in (QED) . Because 3 × δ(p0 − q0) . (20) ¡ ¢ π k2 = 0 6= 0 (16) µν With this manifestly gauge-invariant formulation of QED where πµν is the vacuum polarization tensor, the photon ac- there follows simple proof of infrared exponentiation. quires a nonzero mass. In qcd, due to the self-coupling of the gluons, the gluon In the causal theory there is no regularization ambiguity field must be dressed too. This is an open problem which for the photon mass [4], in contrast to the ambiguities found affects the whole structure of qcd. Indeed, the “ellipsoi- by Deser, Jackiw and Templeton in [9]. dal bound” on the Gribov horizon contradicts the asymptotic As remarked the true scattering matrix is recovered only behaviour at large momentum predicted by the perturbative in the adiabatic limit g → 1 which was shown in [7] for renormalization group [21]. The relation between the infra- massive theories, and in [10] for QED (see also [11] for an red problem and gauge invariance remains thus obscure in approach in terms of the algebra of observables). In QED these theories, and the limit g → 1 remains open. there exists the Bloch-Nordsieck cancellation, but the latter In addition to ultraviolet and infrared divergences, there is not effective in non abelian gauge theories. For theories arises an independent divergence in quantum field theory, such as QED, if we call Fn the amplitude for a given pro- that of the (already renormalized) S-matrix series cess, with the additional production of n (“soft”) photons X∞ (“Brehmsstrahlung”), the overall (inclusive) physical proba- S(g = α) = S αn . (21) bility n X n=0 2 P = |Fn| (17) n We review this aspect briefly following [15]. Typically, L is infrared finite. In QED we also have exponentiation of Sn ∼ (n!) for some L (e.g. L = 1, 2,...) due to the growth of the number of graphs at each order n . infrared divergences: in any S-matrix amplitude F0 , the infrared divergences may be factored out, Thus, S(α) is a divergent series, but which is believed to be asymptotic, e.g., for the gyromagnetic ratio (replacing D F0 = e G0 , (18) S(α)): µ ¶ where D is a simple explicit divergent quantity, and G a g − 2 1 α ³α´2 0 (α) = − 0.328 + 0(α6) new pertubative series, which is free of infrared divergences. 2 2 π π 1346 Walter F. Wreszinski

XN where n an(x − a) is asymptotic to f of order N at 1 α = fine structure constant = . (22) n=0 137.0 ... a for all N = 0, 1, 2,... then one says that the series X∞ n Given a function f defined on an open interval a < x < b , an(x − a) is asymptotic to f at a and writes one says that the function fN defined on the interval, is n=0 asymptotic to f of order N at a if ( x → a+ means the limit from the right): X∞ n f(x) ∼ an(x − a) . (25) f(x) − fN (x) x=0 lim N = 0 . (23) x→a+ (x − a) −N Now, if lim [f(x) − fN (x)] [x − a] = 0 , it evi- This general definition applies in particular to the expression x→a −k dently follows that lim [f(x) − fN (x)] [x − a] = 0 XN x−a f (x) = a (x − a)n . (24) XN N n n n=0 for k = 0, 1,...,N − 1 , so when an(x − a) is n=0 If one is given a sequence {a0, a1, a2,...} and asymptotic to f of order N at a , then:

lim f(x) = a0 − f(a+), (26) x→a+

−1 0 lim [f(x) − f(a+)][x − a] = a1 = f (a+), (27) x→a+ . . " # NX−1 (n) (N) f (a+) n −N f (a+) lim f(x) − (x − a) [x − a] = aN = . (28) x→a+ n! N! n=0

Thus, when a function f is asymptotic of order N at a to N X N−1 n X S(n)(0+) an expression an(x − a) , the function f necessarily n S(α) − SN−1(α) = S(α) − α = n=0 n! has N derivatives from the right at a and the coefficients n=0 an are uniquely determined as Taylor coefficients α Z (N) S (t) N−1 f n(a+) = RN (α); RN (α) = (α − t) dt . a = , n = 0, 1,...,N. (29) (N − 1)! n n! 0 Now RN (α) −→ ∞ for any α 6= 0 but RN (α) ' N→∞ Thus, the assertion that the gyromagnetic anomaly N µ ¶ cN!α . For α small enough we may consider (assuming g − 2 X∞ (α) has the asymptotic series an (α/π)n N large enough – Stirling’s formula; treating N as conti- 2 nuous parameter; justified a posteriori): n=1µ ¶ g − 2 at zero means no more and no less than that (α) ln RN (α) ∼ N ln N + N ln α, 2 is defined for α in some interval a < α < α and has d 0 ln R (α) =∼ 1 + ln N + ln α = 0 derivatives of all orders from the right at 0 . dN N −2 Now, for QED, α ≈ 10 . Suppose now that Sn ' =⇒ ln N = −1 − ln α =⇒ N = e−1e−ln α NX−1 n cn! c > 0 . Consider SN−1(α) = Snα asymp- ' e3 ' 25 for α ≈ 10−2, n=0 totic to S(α) at α = 0 . Thus, by Taylor’s formula with d2 1 ln R (α) = > 0 (minimum!). remainder dN 2 N N Brazilian Journal of Physics, vol. 34, no. 4A, December, 2004 1347

If, however, α ' 10 (qcd) there is no minimum, so that, formally for g ∈ D(R2), even if the series is asymptotic, there is no hope of approxi- Z mating the true function by a finite sum of the perturbation 4 Vg(t) ≡ dx g(x, t): φ (x, 0) : , (30) series! This is the problem of large coupling. It can only be H (t) ≡ H + V (t) , (31) attacked by a construction of the theory in an intrinsically g 0 g nonperturbative way. A first step in this direction is repor- e ted in section 2.1. H(t) ≡ Hg(t) + M1 . (32) An alternative approach would be to to try to prove a (M a constant introduced to make He(t) a positive ope- property of the nonperturbative solution which guarantees rator). Then (W.W., L.A. Manzoni, O. Bolina, preprint, that a special summability method (e.g. Borel’s) applied to Univ. Sao˜ Paulo, to appear in J. Math. Phys.): the perturbation series necessarily yields a unique answer – THEOREM There exists a unique solution of the evolu- that is, the right one, see [15] for a nice review. tion equation A second nonperturbative issue is the triviality of the S matrix. It is conjectured [16] that the (: φ4 :) or (QED) ∂u(t, s) 4 4 i ψ = He(t)U(t, s)ψ , (33) theories, being not asymptotically free [16], lead to a trivial ∂t S matrix, i.e., S = 1 . This is sometimes taken as an argu- e ment in favor of “effective theories”: since the photon has a where ψ ∈ D(H(s)) is dense in F . Further, defining the 1 “Dirac picture propagator” by hadronic component with a probability α ∼ ∼ 1%, 137 D i(H +M)t −i(H +M) QED is not an isolated field of physics but is intimately U (t, s) ≡ e 0 U(t, s)e 0 s , (34) linked to strong interactions. This is physically clear, but, simply theoretically, what is then the renormalized (Dyson) then (s − lim means the strong limit, i.e., on every vector φ S-matrix of (QED)4? of F ), S(g) ≡ s − limU D(t, s) , (35) It cannot be asymptotic if it is trivial, and yet it yields t→+∞ s→−∞ the best and most precise results of science! Weinberg [16] argues that the rigorous lattice-field theory results of exists and satisfies the unitary a) and first causality condi- Aizenman-Frohlich¨ are not conclusive to show triviality of tion b) (for supp g1 > supp g2 ). 4 The second causality condition and Lorentz covariance (: φ :)4 , as well as that the renormalization-group argu- 4 are still open, as well as the adiabatic limit along the lines of ments are inconclusive to show triviality of (: φ :)4 and [7] or [11]. (QED)4. Nevertheless triviality might be true, at least for 4 (: φ :)4 ! A partial result in this direction is in section 2.2. 2.2 Triviality of the S Matrix There are virtually no results on the issue of triviality of the S matrix. K Hepp [17] studied some models which have 2 Nonperturbative Results and Pro- no vacuum polarization (a persistent vacuum), but nontrivial blems renormalization of the mass and the coupling constant. He called these Lee models, and this class of (partially soluble) models the “Royaume Intermediaire”.´ 2.1 Nonperturbative construction of S(g) Let H0 denote the free Hamiltonian of scalar Bosons of (with L.A. Manzoni and O. Bolina) mass m > 0, s Z B² = [−π/², π/²] , with ² > 0 ; (36) ∗ Let H0 = ω(k)a (k)a(k)dk , with ω(k) = ²−1 plays the role of a (momentum) cutoff, which later tends ¡ ¢1/2 k2 + m2 , be the free field Hamiltonian corresponding to infinity in an appropriate manner ([17], [18]). The Hamil- to a zero-time scalar field φ(x, 0) of mass m > 0 , and, tonian is:

H² = H0 + V (37) where Z Z Y4 d~p V = λ √ i δ (~p + ~p − ~p − ~p ) a∗(~p )a∗(~p )a(~p )a(~p ), (38) ² µ 1 2 3 4 1 2 3 4 B² i=1 i 1348 Walter F. Wreszinski and q 2 2 µi = µ(~pi) = ~p i + m , m > 0 .

The number operator commutes with H and Fock space [4] G. Scharf, W.F. Wreszinski, B.M. Pimentel, and J.L. decomposes into (labelled by N ) dynamically independent Tomazelli, Ann. Phys. 231, 185 (1994). sectors, with no vacuum polarization (but: renormalization [5] N. Nakanishi and I. Ojima, Covariant Operator Forma- of mass and coupling constant are necessary!) lism of Gauge Theories and Quantum Gravity, World There are the following results [18]; s is given by (36)! Scientiﬁc Lect. Notes in Phys. vol. 27. THEOREM [18] [6] M. Dutsch¨ and B. Schroer, J. Phys, A33, 4317 (2000). a) If s = 3, either the theory is trivial or E0(N) ≤ [7] H. Epstein and V. Glaser in Renormalization Theory, 2 −BN , with B > 0 independent of N , and E0(N) the ed. by G. Velo and A.S. Wightman, pp. 193–254 ground state energy in the sector of N particles (thus, in this (1976). case the theory is physically inacceptable, because it leads [8] S. Elitzur, Phys. Rev. D12, 3978 (1975). to inﬁnite energy density); [9] S. Deser, R. Jackiw, and S. Templeton, Ann. Phys. b) If s = 2 , the theory is nontrivial. 140, 372 (1982). To the extent that (37), (38) is an approximation of [10] Ph. Blanchard and R. Seneor,´ Ann. Inst. Henri Poin- 4 (: φ :)s+1 (discussed in [18]), the previous theorem shows care´ A23, 147 (1975). the marked difference between s = 2 (s + 1 = 3) and [11] M. Dutsch¨ and K. Fredenhagen, Comm. Math. Phys. s = 3 (s + 1 = 4) , which seems to support the triviality 203, 71 (1999). conjecture. Much remains to be done concerning this im- [12] R. Doria, J. Frenkel, and J.C. Taylor, Nucl. Phys. portant issue! B168, 73 (1980). [13] M. Bergere´ and L. Szymanovski, Phys. Rev. D26, 3350 (1982). Acknowledgement [14] D. Rello, Phys. Rev. D29, 2282 (1984). [15] A.S. Wightman, Should We Believe in Quantum Field The author is grateful to Prof. P. van Nieuwenhuizen Theory? Cargese´ Lectures, 1984. for raising several questions on the causal theory, as well as [16] S. Weinberg, The Quantum Theory of Fields, vol II - instructive comments. Cambridge University Press 1996. [17] K. Hepp, Theorie´ de la Renormalisation´ Springer REFERENCES 1969. [18] F.A.B. Coutinho, J.F. Perez, and W.F. Wreszinski, [1] H. Epstein and V. Glaser, Ann. Inst. Henri Poincare´ Ann. Phys. 277, 94 (1999). A19, 211 (1973). [19] M. Requardt, Comm. Math. Phys. 50, 259 (1976). [2] G. Scharf, Finite Quantum Electrodynamics: the Cau- [20] M. Dutsch,¨ Inst. J. Mod. Phys. A12, 3205 (1997). sal Approach Second edition - Springer, 1995. [21] G. F. Dell’Antonio and D. Zwanziger, Nucl. Phys. B [3] G. Scharf, Quantum Gauge Theories, A True Ghost 326, 333 (1989). Story - Wiley 2001.