aa r X i v : . [ h e p - t h ] J un Stability of QED
M. P. Fry
School of Mathematics, University of Dublin, Trinity College, Dublin 2, Ireland (Dated: November 20, 2018)It is shown for a class of random, time-independent, square-integrable, three-dimensional mag-netic fields that the one-loop effective fermion action of four-dimensional QED increases faster thana quadratic in B in the strong coupling limit. The limit is universal. The result relies on theparamagnetism of charged spin - 1 / PACS numbers: 12.20.Ds, 11.10.Kk, 11.15.Tk
I. INTRODUCTION
Integrating out the fermion fields in four-dimensionalQED continued to the Euclidean metric results in themeasure for the gauge field integration dµ ( A ) = Z − e − R d x ( F µν F µν +gauge fixing) × det ren (1 − eS A ) Y x,µ dA µ ( x ) , (1.1)where det ren is the renormalized fermion determinant de-fined in Sec. II; S is the free fermion propagator, and Z is chosen so that R dµ ( A ) = 1. In the limit e = 0 theGaussian measure for the potential A µ is chosen to havemean zero and covariance Z dµ ( A ) A µ ( x ) A ν ( y ) = D µν ( x − y ) , (1.2)where D µν is the free photon propagator in some fixedgauge. Naively, integration over the fermion fields pro-duces the ratio of determinants det( P + e A + m ) / det( P + m )which is not well-defined; det ren makes sense of this ra-tio. It is gauge invariant and depends only on the fieldstrength F µν and invariants formed from it.We have chosen to introduce this paper with an abruptintrusion of definitions in order to emphasize the centralrole of det ren in QED: it is everywhere. It is the origin ofall fermion loops in QED. If there are multiple chargedfermions then det ren is replaced by a product of renor-malization determinants, one for each species. For ourpurpose here it is sufficient to consider one fermion.The nonperturbative calculation of det ren reduces tofinding the eigenvalues of S A , Z d yS ( x − y ) A ( y ) ψ n ( y ) = 1 e n ψ n ( x ) . (1.3)There are at least two complications. Firstly, S A is nota self-adjoint operator, and so many powerful theoremsfrom analysis do not apply. And secondly, since A µ ispart of a functional measure, it is a random field, makingthe task of calculating the e n for all admissible fieldsimpossible. What can be done is to expand ln det ren , the one-loop effective action, in a power series in e . Thenthe functional integration can be done term-by-term toobtain textbook QED.The first nonperturbative calculation of det ren wasdone by Heisenberg and Euler [1] seventy five years agofor the special case of constant electric and magneticfields. Their paper gave rise to a vast subfield knownas quantum field theory under the influence of externalconditions. A comprehensive review of this body of workrelevant to det ren is given by Dunne [2].An outstanding problem is the strong field behavior ofdet ren that goes beyond constant fields or slowly varyingfields or special fields rapidly varying in one variable [2,3]. That is, what is the strong field behavior of det ren for a class of random fields F µν on R ? What if ln det ren increases faster than a quadratic in F µν for such fields? Isdet ren integrable for any Gaussian measure in this case?This is a question with profound implications for the sta-bility of QED in isolation. Of course, QED is part ofthe standard model, thereby making the overall stabil-ity question a much more intricate one. Nevertheless,the stability of QED in isolation remains unknown anddeserves an answer.In this paper we consider the case of square inte-grable, time-independent magnetic fields B ( x ) defined on R . There are additional technical conditions on B in-troduced later. The magnetic field lines are typicallytwisted, tangled loops. We find thatlim e →∞ ln det ren e ln e = || B || T π , (1.4)where || B || = R d x B · B ( x ), and T is the size of thetime box. Since e always multiplies B , this means thatln det ren is growing faster than a quadratic in B . In the We note here progress in scalar QED since the review [2] ingoing beyond these fields. Using the multidimensional worldlineinstanton technique the vacuum pair production rate has beencalculated from the one loop effective action of a charged scalarparticle in selected two and three-dimensional electric fields [4].These fields have to be sufficiently regular in order to define aformal functional semiclassical expansion of the quantum me-chanical path integral representation of the effective action. Theextension of this technique to spinor QED has not been done yet. constant field case this result is formally equivalent to theHeisenberg-Euler result [1] and to calculations relatingthe effective Lagrangian to the short-distance behaviorof QED via its perturbative β -function [2]. What is no-table here is that the strong coupling limit of ln det ren isuniversal.To achieve universality the derivation of (1.4) mustrely on general principles. One of these is the con-jectured ”diamagnetic” inequality for Euclidean three-dimensional QED, namely (cid:12)(cid:12) det QED (1 − eS A ) (cid:12)(cid:12) ≤ . (1.5)The fermion determinant in (1.5) is defined in Sec.II.The diamagnetic inequality is known to be true for latticeformulations of QED obeying reflection positivity andusing Wilson fermions [5–7]. Since Wilson fermions areCP invariant there is no Chern-Simons term to interferewith the uniqueness of det QED [8]. And since det QED isgauge invariant there are no divergences when the latticespacing for the fermions is sent to zero. As stated bySeiler [7], (1.5) is more an obvious truth than a conjec-ture.Since det QED (cid:12)(cid:12) e =0 = 1 and det QED has no zeros in e for real values of e when m = 0 [9], (1.5) can be rewrittenas 0 < det QED ≤ . (1.6)An inspection of Eq.(2.4) below indicates that (1.6) is areflection of the tendency of an external magnetic fieldto lower the energy of a charged fermion. Therefore, thehistoric heading of (1.5) and (1.6) as ”diamagnetic” in-equalities is a misnomer; paramagnetic inequalities wouldbe a more accurate designation. The detailed justifica-tion for going from (1.5) to (1.6) is given in Sec.II.The second general principle underlying (1.4) is thediamagnetism of charged spin-0 bosons in an externalmagnetic field. This is encapsulated in one of the versionsof Kato’s inequality discussed in Sec. III.The final essential input to (1.4) is a restriction onthe class of fields needed to obtain the limit. These re-strictions are summarized in Sec. IV. As the foregoingremarks indicate, QED is central to the derivation of(1.4), and it is to the connection between QED andQED that we now turn. II. QED AND QED A. The connection
The connection has been dealt with previously [10]. Inorder to make this paper reasonably self-contained wewill review the relevant definitions and results. The up-per bound on det ren obtained in [10] is not optimal; itwill be optimized here. The renormalized and regularized fermion determinantin Wick-rotated Euclidean QED with on-shell renormal-ization, det ren , may be defined by Schwinger’s propertime representation [11]ln det ren (1 − eS A ) = 12 ∞ Z dtt (cid:16) Tr n e − P t − exp h − ( D + e σ µν F µν ) t i(cid:27) + e || F || π (cid:19) e − tm , (2.1)where D µ = P µ − eA µ , σ µν = (1 / i )[ γ µ , γ ν ] , γ † µ = − γ µ , || F || = Z d xF µν ( x ), and e is assumed to be real.We choose the chiral representation of the γ -matrices sothat σ ij = (cid:18) − σ k − σ k (cid:19) , i, j, k = 1 , , A µ = (0 , A ( x )) with x in R . Then (2.1) reducesto ln det ren = T ∞ Z dtt (cid:20) πt ) / Tr (cid:16) e − P t − exp (cid:8) − [( P − e A ) − e σ · B ] t (cid:9)(cid:19) + e || B || π (cid:21) e − tm , (2.2)where T is the dimension of the time box, and the factor2 is from the partial spin trace. Clearly we must have B ∈ L ( R ). If A is assumed to be in the Coulomb gauge ∇ · A = 0, then by the Sobolev-Talenti-Aubin inequality[12] Z d x B ( x ) · B ( x ) ≥ (cid:18) π (cid:19) / X i =1 (cid:18)Z d x | A i ( x ) | (cid:19) / . (2.3)So we must also have A ∈ L (IR ).In analogy with det ren in (2.1), without the chargerenormalization subtraction, det QED may be defined byln det QED ( m ) = 12 ∞ Z dtt Tr (cid:16) e − P t − exp (cid:8) − [( P − e A ) − e σ · B ] t (cid:9)(cid:19) e − tm . (2.4)This definition and regularization of det QED is parityconserving and gives no Chern-Simons term. Substi-tuting (2.4) in (2.2) and, noting that π − ∞ R dE e − tE =(4 πt ) − / , we obtain [10]ln det ren = 2 Tπ ∞ Z dE (cid:18) ln det QED ( E + m )+ e || B || π / ∞ Z dtt / e − ( E + m ) t (cid:19) = Tπ ∞ Z m dM √ M − m (cid:18) ln det QED ( M ) + e || B || π √ M (cid:19) . (2.5)Result (2.5) will be referred to repeatedly in what follows. B. Justification of (1.6)
Continuing our review of previous work we turn to thederivation of the upper bound on ln det ren in (1.4). Sincethe degrees of divergence of the first, second and third-order contributions to ln det
QED are 2,1 and 0, respec-tively, these must be dealt with separately. Their def-inition is obtained from the expansion of (2.4) through O ( e ), resulting inln det QED (1 − eS A ) = − e π Z d k (2 π ) | ˆ B ( k ) | × Z dz z (1 − z )[ z (1 − z ) k + m ] / + ln det (1 − eS A ) , (2.6)where ln det defines the remainder and ˆ B is the Fouriertransform of B . Definition (2.4) assigns the value of zeroto the terms of order e and e . The argument of det QED has been changed to indicate its origin as the formal ratioof QED determinants det( P − e A + m ) / det( P + m ).Note the minus sign in (2.6) pointing to paramagnetism.The following theorems are essential for what follows:Theorem 1 [6, 13, 14]. Let the operator S A in det be transformed by a similarity transformation to K =( p + m ) / S A ( p + m ) − / . This leaves the eigenval-ues of S A invariant. Then K is a bounded operator on L ( R , d x ; C ) for A ∈ L p ( R ) for p >
3. Moreover, K is a compact operator belonging to the trace ideal I p , p > I p (1 ≤ p < ∞ ) is defined as thosecompact operators A with || A || pp = Tr(( A † A ) p/ ) < ∞ .From this it follows that the eigenvalues 1 /e n of S A obtained from (1.3) specialized to three dimensions areof finite multiplicity and satisfy ∞ P n =1 | e n | − p < ∞ for p >
3. The eigenfunctions ψ n belong to the Sobolev space L ( R , √ k + m d k : C ). None of the e n are real for m = 0 [9].Theorem 2 [15–17]. Define the regularized determinant det n (1 + A ) = det " (1 + A ) exp n − X k =1 ( − k A k /k ! . (2.7)Then det n can be expressed in terms of the eigenvaluesof A ∈ I p for n ≥ p .Accordingly, det in (2.6) is defined and can be repre-sented as [17]det (1 − eS A ) = ∞ Y n =1 "(cid:18) − ee n (cid:19) exp X k =1 (cid:18) ee n (cid:19) k /k ! . (2.8)The reality of det for real e and C -invariance requirethat the eigenvalues e n appear in the complex plane asquartets ± e n , ± e ∗ n or as imaginary pairs when m = 0. Asexpected, the expansion of ln det in powers of e beginsin fourth order.We have established that det | e =0 = 1 and that det has no zeros for real values of e . Therefore, by (2.6)det QED > e , thereby allowing one to gofrom (1.5) to (1.6). It might be objected that this isobvious, but we will need the detailed information intro-duced about det in the sequel.The determinant det is an entire function of e consid-ered as a complex variable, meaning that it is holomor-phic in the entire complex e -plane. Since ∞ P n =1 | e n | − − ǫ < ∞ for ǫ >
0, its order is at most 3 [16, 18]. This meansthat for any complex value of e , and positive constants A, K, | det | < A ( ǫ ) exp( K ( ǫ ) | e | ǫ ) for any ǫ > e ln det ≤ e π Z d k (2 π ) | ˆ B ( k ) | Z dz z (1 − z )[ z (1 − z ) k + m ] / . (2.9)This is a truly remarkable inequality. Referring to (2.9),det ’s growth is slower on the real e -axis than its poten-tial growth in other directions. We also note that det is largely unknown. Even the reduction of the fourth-order term in its expansion to an explicitly gauge invari-ant form involving only B -fields requires a huge effortwhen the fields are not constant [19]. The sixth-orderreduction has not been completed as far as the authorknows. C. Upper bound on det ren
Insert (2.6) in (2.5) and getln det ren = e T π Z d k (2 π ) | ˆ B ( k ) | ∞ Z dz z (1 − z ) × ln (cid:20) z (1 − z ) k + m m (cid:21) + Tπ ∞ Z m dM √ M − m ln det ( M ) . (2.10)The objective here is to obtain the behavior of ln det ren when the coupling e is large, real and positive. Since e always multiplies B we introduce the scale parameter B = max x | B | , which has the dimension of M . Why B is finite will be explained in Sec. III.B. Then the integralin (2.10) is broken up into e B R m and ∞ R e B .Substitution of (2.9) into the lower range integral givesln det ren ≤ e T π Z d k (2 π ) | ˆ B ( k ) | Z dz z (1 − z ) × ln (cid:18) e B + 2 z (1 − z ) k − m m (cid:19) + Tπ ∞ Z e B dM √ M − m ln det ( e B , M ) . (2.11)We have simplified the argument of the logarithm using2 √ xy ≤ x + y for x, y ≥
0. Then for e B ≫ m ln det ren ≤ e T || B || π ln (cid:18) e B m (cid:19) + Tπ ∞ Z e B dM √ M − m ln det ( e B , M )+ O (cid:18) eT R d x B · ∇ B B (cid:19) . (2.12)The integral in (2.12) can be estimated by making a largemass expansion of ln det . This is facilitated by inserting(2.6) in (2.4) and examining the small t region of ln det ’sresulting proper time representation. The details of thisexpansion are in Sec. 3B of [10], and give the resultln det ( e B , M ) = M →∞ ∞ Z dtt (4 πt ) − / e − tM × Z d x (cid:20) e t ( B · B ) + O ( e t B · BB · ∇ B ) (cid:21) = e R ( B · B ) πM + O (cid:18) e R B · BB · ∇ B M (cid:19) . (2.13)In the first line of (2.13) it is assumed that the heatkernel expansion is an asymptotic expansion in t in thestrict sense of its definition, namely [20] < x | e − t [( P − e A ) − e σ · B ] | x > − (4 πt ) − / N X n =0 a n ( x ) t n g t → (4 πt ) − / a N +1 ( x ) t N +1 . (2.14)This must hold for every N . A necessary condition for(2.14) is that B be infinitely differentiable to ensure thateach coefficient a n is finite. As far as the author knows itis not known yet if this is a sufficient condition. So (2.14)is an assumption that may require additional conditionson B . Only coefficients a n of O ( e n ) , n ≥ ’s expansion.The t-integration in (2.13), although extending to in-finity, is limited to small t since M → ∞ due to theparameter e B in (2.12). Substituting (2.13) in (2.12)results inln det ren ≤ e || B || T π ln (cid:18) e B m (cid:19) + O (cid:18) e T R ( B · B ) B (cid:19) + O (cid:18) eT R B · ∇ B B (cid:19) , (2.15)or lim e →∞ ln det ren e ln e ≤ || B || T π , (2.16)consistent with (1.4). This bound is independent of thecharge renormalization subtraction point. If the subtrac-tion were made at photon momentum k = µ instead of k = 0 then the ln m terms in (2.10) and (2.11) wouldbe replaced with ln[ z (1 − z ) µ + m ], which has nothingto do with strong coupling.The scaling procedure used here is designed to obtainthe least upper bound on ln det ren . In [10] we chose tobreak up the M -integral as e || B || R m and ∞ R e || B || . This re-sulted in a fast 1 /e falloff of the ln det terms comparedto e here, but gave a weaker upper bound on ln det ren ,namely lim e →∞ ln det ren e ln e ≤ || B || T π . (2.17)We mention that the coefficient 1/960 in (3.16) in [10]should be 1/360.Here we might have chosen a more general scaling suchas e α (ln e ) β B or e α (ln ln e ) β B , etc., with α ≥ , β > α || B || T / π . The case α < e B is an optimal one. III. LOWER BOUND ON det ren
A. Fundamentals
On referring to (2.5) the lower bound on det ren willcome from operations on ln det
QED . We begin withthe operator identity (A2) in Appendix A applied toln det QED in (2.4). Letting X = ( P − e A ) and Y = − e σ · B we obtainln det QED = − ∞ Z dtt Tr e t Z ds e − ( t − s )( P − e A ) × σ · B e − s ( P − e A ) + e t Z ds t − s Z ds e − ( t − s − s )[( P − e A ) − e σ · B ] × σ · B e − s ( P − e A ) σ · B e − s ( P − e A ) ! e − m t + 12 ∞ Z dtt Tr (cid:16) e − P t − e − ( P − e A ) t (cid:17) e − tm . (3.1)The spin trace in the first term is zero, and the last term,after tracing over spin, is the one-loop effective action ofscalar QED ,ln det S QED = ∞ Z dtt e − tm Tr (cid:16) e − P t − e − ( P − e A ) t (cid:17) . (3.2)Thus,ln det QED = ln det S QED − e ∞ Z dtt e − tm Tr t Z ds × t − s Z ds e − ( t − s − s )[( P − e A ) − e σ · B ] × σ · B e − s ( P − e A ) σ · B e − s ( P − e A ) (cid:19) , (3.3)remembering that the factor 1 / A = [( P − e A ) + m ] − . In Appendix B it isshown that ∆ / A σ · B ∆ / A ∈ I ; that is, it is a Hilbert-Schmidt operator provided B ∈ L and m = 0. Then(2.7) givesln det (1 − e ∆ / A σ · B ∆ / A )= ln det h (1 − e ∆ / A σ · B ∆ / A ) e e ∆ / A σ · B ∆ / A i = Tr ln h (1 − e ∆ / A σ · B ∆ / A ) e e ∆ / A σ · B ∆ / A i = Tr ∞ Z dtt e − tm (cid:16) e − ( P − e A ) t − e − [( P − e A ) − e σ · B ] t (cid:17) + e ∆ A σ · B = − e ∞ Z dtt e − tm t Z ds t − s Z ds Tr (cid:16) e − ( t − s − s )[( P − e A ) − e σ · B ] × σ · B e − s ( P − e A ) σ · B e − s ( P − e A ) (cid:19) . (3.4) In going from the penultimate to the last line in (3.4) usewas again made of the identity (A2). Substituting (3.4)in (3.3) givesln det QED = 12 ln det (1 − e ∆ / A σ · B ∆ / A )+ln det S QED . (3.5)As ln det QED and ln det are well-defined by our choiceof fields, so is ln det SQED in (3.5). What has been accom-plished here is to isolate the Zeeman term σ · B in ln det .Since ∆ / A σ · B ∆ / A is Hilbert-Schmidt and self-adjoint,ln det is susceptible to extensive analytic analysis.Substitute (3.5) in (2.5):ln det ren = Tπ ∞ Z m dM √ M − m × (cid:18)
12 ln det (1 − e ∆ / A σ · B ∆ / A )+ ln det S QED + e || B || π √ M (cid:19) . (3.6)We now introduce two central inequalities. The first re-lies on the diamagnetism of charged scalar bosons as ex-pressed by Kato’s inequality in the form [21, 22]Tr (cid:16) e − ( P − e A ) t (cid:17) ≤ Tr e − P t . (3.7)This implies that on average the energy eigenvalues ofsuch bosons rise in a magnetic field and hence by(3.2)that [22] ln det S QED ≥ . (3.8)The second inequality is introduced beginning with thepenultimate line of (3.4). Noting that the spin trace ofthe ∆ A σ · B term is zero, thenln det (cid:16) − e ∆ / A σ · B ∆ / A (cid:17) = ∞ Z dtt e − tm × Tr (cid:16) e − t ( P − e A ) − e − [( P − e A ) − e σ · B ] t (cid:17) . (3.9)By the Bogoliubov-Peierls inequality [23, 24] and Sec.2.1,8 of [25]Tr e − [( P − e A ) − e σ · B ] t ≥ Tr e − t ( P − e A ) t e − te< σ · B > , (3.10)where < σ · B > = Tr (cid:16) σ · B e − t ( P − e A ) t (cid:17) Tr e − ( P − e A ) t = 0 . (3.11)Hence, ln det (cid:16) − e ∆ / A σ · B ∆ / A (cid:17) ≤ . (3.12)consistent with (3.5) when combined with (1.6) and (3.8).There is another reason why (3.12) holds. Let C = e ∆ / A σ · B ∆ / A . Since C is Hilbert-Schmidt,ln det (1 − C ) = ln det (cid:2) (1 − C ) e C (cid:3) = Tr [ln(1 − C ) + C ]= 12 Tr ln(1 − C )= 12 ∞ X n =1 ln(1 − λ n ) . (3.13)The third line of (3.13) follows from the second sincethe trace over spin eliminates all odd powers of C . In thelast line we introduced the real eigenvalues λ n of e ∆ / A σ · B ∆ / A . Since ln det is real and finite then | λ n | < n , giving (3.12). Because ∆ / A σ · B ∆ / A ∈ I , it isa compact operator, and so the λ n are countable and offinite multiplicity.Now consider ∂∂m ln det ( m ) = ∞ Z dt e − tm × Tr (cid:16) e − [( P − e A ) − e σ · B ] t − e − ( P − e A ) t (cid:17) ≥ , (3.14)by (3.9)-(3.11). Therefore, det is a monotonically in-creasing function of m .Next, break up the M -integral in (3.6) as in Sec.II.C:ln det ren = Tπ e B Z m dM √ M − m (cid:18)
12 ln det (1 − e ∆ / A σ · B ∆ / A )+ ln det S QED + e || B || π √ M (cid:19) + Tπ ∞ Z e B dM √ M − m (cid:18) ln det QED + e || B || π √ M (cid:19) , (3.15)where we reinserted (3.5) into the upper-range M -integral. By (3.14) e B Z m dM √ M − m ln det ( M ) ≥ ln det (cid:12)(cid:12)(cid:12)(cid:12) M m e B Z m dM √ M − m = 2 ln det (cid:16) − e ∆ / A σ · B ∆ / A (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) M m p e B − m . (3.16)Hence, (3.8) and (3.16) result in (3.15) becomingln det ren ≥ Tπ p e B − m ln det (cid:16) − e ∆ / A σ · B ∆ / A (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) M m + e T || B || π ln (cid:18) e B m (cid:19) + e T π || B || ln r − m e B ! + Tπ ∞ Z e B dM √ M − m (cid:18) ln det QED + e || B || π √ M (cid:19) . (3.17)We now turn to the strong coupling behavior of ln det . B. Strong coupling behavior of ln det The eigenvalues λ n in (3.13) are obtained from e ∆ / A σ · B ∆ / A ϕ n = λ n ϕ n , (3.18)for ϕ n ∈ L following the remark under (B3) in AppendixB. Letting ∆ / A ϕ n = ψ n gives (cid:20) ( P − e A ) − e σ · B λ n (cid:21) ψ n = − m ψ n , (3.19)where ψ n ∈ L provided m = 0. This follows from(B5) and Young’s inequality (B7). The requirement that m = 0 follows from the role of the eigenvalues { λ n } ∞ n =1 as adjustable coupling constants whose discrete valuesresult in bound states with energy − m for a fixed valueof e . Since the operator ( P − e A ) − e σ · B ≥
0, suchbound states are impossible unless | λ n | < n ,which is the physical reason why (3.12) is true. Inspec-tion of (3.19) suggests that as e increases | λ n | likewiseincreases for fixed n to maintain the bound state energyat − m . This is illustrated by the constant field casethat is excluded from our analysis: | λ n | = | eB | (2 n + 1) | eB | + m , n = 0 , , . . . . (3.20)Because the operator ∆ / A σ · B ∆ / A is Hilbert-Schmidtthe eigenfunction ϕ n has finite multiplicity, and the λ n in (3.13) are counted up to this multiplicity. To estimatethe multiplicity note that the eigenfunctions ϕ n and ψ n are in one-to-one correspondence. Next, note that for ψ ∈ L ( R ; C ) and a generic λ with | λ | < ψ, [( P − e A ) − eλ σ · B ] ψ ) ≥ ( ψ, [( P − e A ) − (cid:12)(cid:12)(cid:12) eλ (cid:12)(cid:12)(cid:12) | B | ) ψ. (3.21)Thus the Hamiltonian on the left, H + , dominates that onthe right, H − . Let N − m ( H ) denote the dimension of thespectral projection onto the eigenstates of Hamiltonian H with eigenvalues less than or equal to − m . Because H + ≥ H − then N − m ( H + ) ≤ N − m ( H − ). N − m ( H + )is an overestimate of the number of the bound states of H + at − m for a fixed value of λ but satisfactory for ourpurpose here.By the Cwinkel-Lieb-Rozenblum bound in the form[26] N − m ( H − ) ≤ C Z d x h(cid:12)(cid:12)(cid:12) eλ (cid:12)(cid:12)(cid:12) | B ( x ) | − m i / , (3.22)where [ a ] + = max( a,
0) and C = 2 × . | λ n | = O (1), we are con-fident that the degeneracy/multiplicity associated witheach λ n in (3.13) does not exceed c | e | / R d x | B | / ,where c ≥ . n > N beyondwhich λ n assumes its asymptotic form as discussed be-low. Therefore, for n ≤ N we will estimate the sum in(3.13) by factoring out the common maximal degeneracy c | e | / R d x | B | / and treat each λ n in the factored sumas having multiplicity equal to one. Those λ n , if any,that vanish as e → ∞ give a subdominant contributionto ln det in (3.13) since by inspection their contributiongrows at most as λ n | e/λ n | / .We now turn to the large e dependence of λ n . Fromhere on we assume that ψ n is normalized to one. ByC-invariance we may assume e >
0. Now consider theexpectation value of (3.19): < n | ( P − e A ) | n > − eλ n < n | σ · B | n > = − m . (3.23)From (3.23) if < n | σ · B | n > > λ n > viceversa . Therefore, we need only consider λ n > λ n = (cid:20) < n | ( P − e A ) | n >e < n | σ · B | n > + m e < n | σ · B | n > (cid:21) − , (3.24)where < n | σ · B | n > = 0 as (3.23) must be satisfied. Thecase λ n = 0 for some n corresponding to < n | σ · B | n > = 0can be ignored as λ n = 0 contributes nothing to ln det in (3.13). An easy estimate gives | ( ψ n , σ · B ψ n ) | ≤ ( ψ n , | B | ψ n ) ≤ max x | B ( x ) | . (3.25)Because B ∈ L and is assumed infinitely differentiablethen max x | B | is finite. Hence, < n | σ · B | n > is a boundedfunction of e and n .Now consider the ratio R n = < n | ( P − e A ) | n > /e < n | σ · B | n > in (3.24). The case R n −−→ e ≫ λ n → ∞ . The case R n −−→ e ≫ ∞ implies λ n →
0, which gives a subdominant contribution to (3.13) asdiscussed above. The final possibility is 1 ≤ R n < ∞ for e → ∞ . The case R n → e → ∞ happens if < n | ( P − e A ) | n > ∼ e < n | σ · B | n > . Since ψ n ∈ L , < n | ( P − e A ) − e σ · B | n > = 0 implies σ · ( P − e A ) ψ n = 0.Now this may happen for the B -fields considered so far.But if we exclude zero-mode supporting B fields [27] fromour analysis it cannot. By so doing we can exclude thecase λ n = 1 − δ n ( e ), δ n ( ∞ ) = 0. We will see below whythis is necessary.We proceed to estimate the strong coupling limit ofln det in (3.13). First, consider the sum for n ≤ N . Weneed only consider 0 < | λ n | < e , including e = ∞ as concluded above. Hence, on factoring out the commonmaximal multiplicity of the λ n we getlim e ≫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X n =1 ln(1 − λ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c e / Z d x | B | / , (3.26)where c is a constant and noting again that the eigen-values λ n → λ n → n → ∞ and 1 / ≤ | ln(1 − λ n ) /λ n | ≤ / λ n < / ∞ P n =1 λ n < ∞ . Consider this sum for n > N and indicate the degeneracy factors µ n explicitly: S ≡ ∞ X n>N µ n ( e ) λ n ( e ) . (3.27)We estimated from (3.22) that µ n ≤ c | e/λ n | / R d x | B | / . So S ≤ ce / Z d x | B | / ∞ X n>N no degeneracy | λ n | / < ∞ . (3.28)This implies that for n > N | λ n ( e ) | = C n ( e ) n ǫ , (3.29)where ǫ > C n is a bounded function of n and e withlim e →∞ C n ( e ) < ∞ . Otherwise | λ n | < n cannotbe satisfied. Accordingly, the series in (3.28) is uniformlyconvergent in e by the Weierstrass M-test and solim e →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∞ X n>N ln(1 − λ n ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) /e / ≤ c Z d x | B | / , (3.30)where c is a constant. From (3.13), (3.26) and (3.30) weconcludelim e →∞ (cid:12)(cid:12)(cid:12) ln det (1 − e ∆ / A σ · B ∆ / A ) (cid:12)(cid:12)(cid:12) /e / ≤ c Z d x | B | / , (3.31)where c is another constant.As a check on (3.31) refer to (3.5). For B ∈ L / ( R )we found [10]ln det QED ≥ − Ze / π Z d x | B ( x ) | / , (3.32)for B ( x ) ≥ B ( x ) ≤ x ∈ R . Z is the dimension ofthe remaining space box. We know that ln det ≤ S QED ≥ B fields it is seen that the strong coupling growth of ln det is consistent with (3.32).Finally, if zero mode supporting B fields were al-lowed we would have obtained ln det = e ≫ O ( e / ln ρ ( e )), ρ ( e ) → e ≫
0, since when | λ n | f e ≫ − δ n ( e ), δ n ( ∞ ) =0, the logarithm in (3.13) gives an additional factorln δ n . As will be seen below the limit (1.4) requireslim e →∞ ln det /e / = finite (or zero). C. Strong coupling limit of (3.17)
It remains to estimate the large coupling limit of thelast term in (3.17), I ≡ Tπ ∞ Z e B dM √ M − m (cid:18) ln det QED + e || B || π √ M (cid:19) = Tπ ∞ Z e B dM √ M − m (cid:18) − e π Z d k (2 π ) | ˆ B ( k ) | × Z dz z (1 − z )[ z (1 − z ) k + M ] + e || B || π √ M +ln det (1 − eS A ) (cid:19) , (3.33)where we substituted (2.6) for ln det QED . Calculation ofthe first two terms in (3.33) is straightforward. The lastterm has already been estimated in Sec. II and is givenby the second term in (2.15). Hence, I = O (cid:18) e T R ( B · B ) B (cid:19) + O (cid:18) eT R B · ∇ B B (cid:19) . (3.34)Taking into account (3.31) and (3.34) we obtain from(3.17) lim e →∞ ln det ren e ln e ≥ || B || T π . (3.35)Equations (2.16) and (3.35) therefore establish (1.4). IV. SUMMARY
The two assumptions underlying (1.4) are first thatthe continuum limit of the lattice diamagnetic inequalitycoincides with (1.5), and second that the heat kernel ex-pansion of the Pauli operator in (2.14) is an asymptoticseries. These assumptions can and should be proven orfalsified.In addition, the result (1.4) assumes that the vectorpotential and magnetic field satisfy the following condi-tions: B ∈ L ( R ) to define ln det ren in (2.5) and to ensurethat ∆ / A σ · B ∆ / A ∈ I following Appendix B. In addi-tion B ∈ L / ( R ) in order that the degeneracy estimatein (3.22) is defined. To ensure that the bound in (3.31)holds, zero mode supporting B fields are excluded. Also B must be infinitely differentiable ( C ∞ ) to ensure thatthe expansion coefficients in (2.14) are finite.If A is assumed to be in the Coulomb gauge then by(2.3) A ∈ L ( R ). If B ∈ L / ( R ) then A ∈ L ( R ) bythe Sobolev-Talenti-Aubin inequality [12]. In order to de-fine det QED it is necessary to assume A ∈ L r ( R ) , r >
3, following the discussion under (2.6). If A ∈ L ( R )and L ( R ), then A ∈ L r ( R ) , < r < || f g || r ≤ || f || p || g || q , (4.1)with p − + q − = r − , p, q, r ≥
1. Since B = ∇ × A and B ∈ C ∞ then A ∈ C ∞ .We note that the sample functions A µ ( x ) supportingthe Gaussian measure in (1.2) with probability one arenot C ∞ . It is generally accepted that they belong to S ′ ( R ), the space of tempered distributions. There-fore, we point out here that the C ∞ functions we in-troduced can be related to A µ ∈ S ′ ( R ) by the convo-luted field A Λ µ ( x ) = R d y f Λ ( x − y ) A µ ( y ) ∈ C ∞ , provided f Λ ∈ S ( R ), the functions of rapid decrease. Then theFourier transform of the covariance R dµ ( A ) A Λ µ ( x ) A Λ ν ( y )derived from (1.2) is ˆ D µν ( k ) | ˆ f Λ ( k ) | , where ˆ f Λ ∈ C ∞ .Since QED must be ultraviolet regulated before renor-malizing, ˆ f Λ can serve as the regulator by choosing, forexample, ˆ f Λ = 1 , k ≤ Λ and ˆ f Λ = 0 , k ≥ . So theneed to regulate can serve as a natural way to introduce C ∞ background fields A Λ µ into det ren – but not the restof dµ in (1.1) – and into whatever else one is calculat-ing. This procedure is a generalization of that used inthe two-dimensional Yukawa model [29].Finally, the obvious generalization of (1.4) for an ad-missible class of fields on R islim e →∞ ln det ren e ln e = 148 π Z d x F µν ( x ) . (4.2)There is no chiral anomaly term since F µν falls off faster than 1 / | x | and R d x ˜ F µν F µν = R d x ∂ α ( ǫ αβµν A β F µν ) = 0., where ˜ F µν = ǫ µναβ F αβ .Equation (4.2) remains to be verified.If (1.4) and (4.2) do indeed indicate instability thenthey are yet another reason why QED should not be con-sidered in isolation. APPENDIX A
The operator indentity on which (3.1) is based is ob-tained as follows. Let [30] F t = e − t ( X + Y ) e tX . Then dF t dt = − e − t ( X + Y ) Y e tX . Integrating gives e − t ( X + Y ) − e − tX = − t Z ds e − ( t − s )( X + Y ) Y e − sX , (A1)known as Duhamel’s formula. Iterating once gives therequired identity: e − t ( X + Y ) − e − tX = − t Z ds e − ( t − s ) X Y e − sX + t Z ds t − s Z ds e − ( t − s − s )( X + Y ) Y e − s X Y e − s X . (A2) APPENDIX B
Here we show that the operator K = ∆ / A σ · B ∆ / A ∈ I and hence that K is Hilbert-Schmidt. This follows[14, 15, 28] if and only if K is a bounded operator on L ( R , d x ; C ) having a representation of the form( Kf )( x ) = Z K ( x, y ) f ( y ) d y, f ∈ L , (B1)where K ( x, y ) = < x | ∆ / A σ · B ∆ / A | y >, (B2)and where K ∈ L ( R × R ; d x × d y ). Moreover, || K || = Z |K ( x, y ) | d xd y. (B3)If it can be shown that K ∈ L then it trivially followsthat K maps L into itself. So consider ||K|| L = 2 X i Z d xd yB i ( x )∆ A ( x, y ) B i ( y )∆ A ( y, x ) ≤ X i Z d xd y | B i ( x ) || ∆ A ( x, y ) || B i ( y ) || ∆ A ( y, x ) | . (B4) A form of Kato’s inequality [5, 21, 31] asserts that theinteracting scalar propagator is bounded by the free prop-agator | ∆ A ( x, y ) | ≤ ∆( x − y ) , (B5)where ∆( x ) = (4 π | x | ) − e − m | x | in three dimensions.Then ||K|| L ≤ π X i Z d xd y | B i ( x ) | | x − y | e − m | x − y | | B i ( y ) | . (B6)By Young’s inequality in the form [23] (cid:12)(cid:12)(cid:12)(cid:12)Z d xd yf ( x ) g ( x − y ) h ( y ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ || f || p || g || q || h || r , (B7)where p − + q − + r − = 2 , p, q, r ≥
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