Stability of QED

Stability of QED

School of Mathematics, University of Dublin, Trinity College, Dublin 2, Ireland (Received 5 July 2011; published 19 September 2011)

It is shown for a class of random, time-independent, square-integrable, three-dimensional magnetic fields that the one-loop effective fermion action of four-dimensional QED increases faster than a quadratic inBin the strong coupling limit. The limit is universal. The result relies on the paramagnetism of charged spin-1=2fermions and the diamagnetism of charged scalar bosons.

DOI:10.1103/PhysRevD.84.065021 PACS numbers: 12.20.Ds, 11.10.Kk, 11.15.Tk

I. INTRODUCTION

Integrating out the fermion fields in four-dimensional QED continued to the Euclidean metric results in the measure for the gauge field integration

dðAÞ ¼Z1e

R

d4_xð1=4F

Fþgauge fixingÞ

detrenð1eS6AÞ

Y

x;

dAðxÞ; (1.1)

where det_ren is the renormalized fermion determinant de-fined in Sec.II;Sis the free fermion propagator, andZis chosen so that RdðAÞ ¼1. In the limit e¼0, the Gaussian measure for the potentialA is chosen to have mean zero and covariance

Z

dðAÞAðxÞAðyÞ ¼DðxyÞ; (1.2)

where D is the free photon propagator in some fixed gauge. Naively, integration over the fermion fields pro-duces the ratio of determinants detð6PeA6 þmÞ=

detð6PþmÞ, which is not well defined;detren makes sense of this ratio. It is gauge invariant and depends only on the field strengthFand invariants formed from it.

We have chosen to introduce this paper with an abrupt intrusion of definitions in order to emphasize the central role ofdetrenin QED: it is everywhere. It is the origin of all fermion loops in QED. If there are multiple charged fer-mions, thendetrenis replaced by a product of renormalized determinants, one for each species. For our purpose here, it is sufficient to consider one fermion.

The nonperturbative calculation of detren reduces to finding the eigenvalues ofS6A,

Z

d4ySðxyÞ6AðyÞcnðyÞ ¼ 1

en

cnðxÞ: (1.3)

There are at least two complications. First,S6Ais not a self-adjoint operator, and so many powerful theorems from analysis do not apply. Second, sinceA is part of a func-tional measure, it is a random field, making the task of calculating the en for all admissible fields impossible. What can be done is to expand ln detren, the one-loop effective action, in a power series ine. Then the functional

integration can be done term-by-term to obtain textbook QED.

The first nonperturbative calculation ofdet_renwas done by Heisenberg and Euler [1] 75 years ago for the special case of constant electric and magnetic fields. Their paper gave rise to a vast subfield known as quantum field theory under the influence of external conditions. A comprehen-sive review of this body of work relevant todetrenis given by Dunne [2].

An outstanding problem is the strong field behavior of detren that goes beyond constant fields or slowly varying fields or special fields rapidly varying in one variable [2,3].1 That is, what is the strong field behavior ofdetren for a class of random fields F onR4? What ifln detren increases faster than a quadratic inFfor such fields? Is detren integrable for any Gaussian measure in this case? This is a question with profound implications for the stability of QED in isolation. Of course, QED is part of the standard model, thereby making the overall stability question a much more intricate one. Nevertheless, the stability of QED in isolation remains unknown and de-serves an answer.

In this paper, we consider the case of square-integrable, time-independent magnetic fields BðxÞ defined on R3_. There are additional technical conditions onBintroduced later. The magnetic field lines are typically twisted, tangled loops. We find that

lim e!1

ln detren e2_lne ¼

k_Bk2_T

242 ; (1.4)

where kBk2_¼R_d3_{xBBðxÞ, and T is the size of the} time box. Since e always multiplies B, this means that ln detren is growing faster than a quadratic in B. In the constant field case, this result is formally equivalent to the

1_{We note here progress in scalarQED}

4since the review [2] in

Heisenberg-Euler result [1] and to calculations relating the effective Lagrangian to the short-distance behavior of QED via its perturbative-function [2]. What is notable here is that the strong coupling limit ofln det_ren is universal.

To achieve universality, the derivation of (1.4) must rely on general principles. One of these is the conjectured ‘‘diamagnetic’’ inequality for Euclidean three-dimensional QED, namely

jdet_QED3ð1eS6AÞj 1: (1.5)

The fermion determinant in (1.5) is defined in Sec.II. The diamagnetic inequality is known to be true for lattice formulations of QED₃ obeying reflection positivity and using Wilson fermions [5–7]. Since Wilson fermions are CP invariant, there is no Chern-Simons term to interfere with the uniqueness of detQED3 [8]. And sincedetQED3 is

gauge invariant, there are no divergences when the lattice spacing for the fermions is sent to zero. As stated by Seiler [7], (1.5) is more an obvious truth than a conjecture.

Sincedet_QED3j_e¼0 ¼1anddetQED3has no zeros inefor

real values ofewhenm0[9], (1.5) can be rewritten as

0<det_QED3 1: (1.6)

An inspection of Eq. (2.4) below indicates that (1.6) is a reflection of the tendency of an external magnetic field to lower the energy of a charged fermion. Therefore,

the historic heading of (1.5) and (1.6) as diamagnetic inequalities is a misnomer; paramagnetic inequalities would be a more accurate designation. The detailed justi-fication for going from (1.5) to (1.6) is given in Sec.II.

The second general principle underlying (1.4) is the diamagnetism of charged spin-0 bosons in an external magnetic field. This is encapsulated in one of the versions of Kato’s inequality discussed in Sec.III.

The final essential input to (1.4) is a restriction on the class of fields needed to obtain the limit. These restrictions are summarized in Sec. IV. As the foregoing remarks indicate,QED3 is central to the derivation of (1.4), and it is to the connection betweenQED3andQED4that we now turn.

II.QED3 ANDQED4

A. The connection

The connection has been dealt with previously [10]. In order to make this paper reasonably self-contained, we will review the relevant definitions and results. The upper bound ondet_renobtained in [10] is not optimal; it will be optimized here.

The renormalized and regularized fermion determinant in Wick-rotated Euclidean QED4 with on-shell renormal-ization,detren, may be defined by Schwinger’s proper time representation [11]

det_renð1eS6AÞ ¼1

2

Z1

0 dt

t

Tr

eP2_t exp

ðD2_þe

2FÞt

þe2jjFk2 242

etm2

; (2.1)

whereD¼PeA,¼ ð1=2iÞ½,y¼ ,kFk2¼

R

d4xF2ðxÞ, andeis assumed to be real. We choose the chiral representation of the-matrices so that

_ij¼ k 0

0 k

;

i; j; k¼1;2;3in cyclic order. Since we will consider time-independent magnetic fields, we setA ¼ ð0;AðxÞÞwithxin R3_{. Then (2.1) reduces to}

ln det_ren¼T 2

Z1

0 dt

t

2

ð4tÞ1=2 Trðe P2_t

expf½ð_Pe_AÞ2_e

BtgÞ þe 2_k

Bk2

122

etm2

; (2.2)

where T is the dimension of the time box, and the fac-tor 2 is from the partial spin trace. Clearly, we must have B2L2_ðR3_{Þ. IfAis assumed to be in the Coulomb} gauge rA¼0, then by the Sobolev-Talenti-Aubin in-equality [12]

Z

d3_x

BðxÞ _BðxÞ

274

16

_1=3X3

i¼1

Z

d3_xjA iðxÞj6

₁₌₃

:

(2.3)

So we must also haveA2L6ðR3Þ.

In analogy withdetrenin (2.1), without the charge renor-malization subtraction,detQED3 may be defined by

lndetQED3ðm

2_Þ

¼1 2

Z1

0 dt

t Trðe P2_t

expf½ðPeAÞ2_eBtgÞetm2

:

(2.4)

This definition and regularization ofdet_QED3is parity con-serving and gives no Chern-Simons term. Substituting (2.4) in (2.2) and, noting that1R1₀ dEetE2

ln detren¼

2T

Z1

0 dE

lndetQED3ðE

2_þm2_Þ

þe2kBk2 243=2

Z1

0 dt t1=2e

ðE2_þm2_Þt

¼T

Z1

m2

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p lndet_QED3ðM2_Þþ e2kBk2

24pffiffiffiffiffiffiffiM2

:

(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 the derivation of the upper bound onln det_renin (1.4). Since the degrees of divergence of the first-, second-, and third-order contributions to ln det_QED3 are 2, 1, and 0, respectively, these must be dealt with separately. Their definition is obtained from the expansion of (2.4) throughOðe3Þ, result-ing in

ln detQED3ð1eSA6 Þ

¼ e2 4

Z _d3_k

ð2Þ3j

^

BðkÞj2Z1 0

dz zð1zÞ

½zð1zÞk2_þm21=2

þln det₄ð1eSA6 Þ; (2.6)

where ln det₄ defines the remainder and B^ is the Fourier transform ofB. Definition (2.4) assigns the value of zero to the terms of ordereande3_{. The argument of det}

QED3 has

been changed to indicate its origin as the formal ratio of QED₃ determinants detð6PeA6 þmÞ=detð6PþmÞ. 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 S6A in det₄ be transformed by a similarity transformation to K ¼

ðp2_þm2_Þ1=4_{SA6 ðp}2_þm2_Þ1=4_{. This leaves the} eigenval-ues of S6A invariant. Then K is a bounded operator on L2_ðR3_;d3_x;C2_{Þ for A2L}p_ðR3_{Þ for p >3. Moreover, K} is a compact operator belonging to the trace ideal Ip, p >3.

The trace ideal Ipð1p <1Þ is defined as those compact operators A with kAkpp¼TrððAyAÞp=2Þ<1. From this it follows that the eigenvalues 1=en ofS6A ob-tained from (1.3) specialized to three dimensions are of finite multiplicity and satisfyP1_n¼1jenjp<1forp >3. The eigenfunctions c_n belong to the Sobolev space L2_ðR3_;pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_k2_þm2_d3_{k;CÞ. None of the e}

n are real for m0[9].

Theorem 2 [15–17]. Define the regularized determinant

detnð1þAÞ ¼det

ð1þAÞexpX n1

k¼1

ð1Þk_Ak_=k

: (2.7)

Thendetncan be expressed in terms of the eigenvalues of A2Ipfornp.

Accordingly, det₄ in (2.6) is defined and can be repre-sented as [17]

det₄ð1eS6AÞ ¼Y 1

n¼1

1 e

en

expX

3

k¼1

e en

_k

=k

:

(2.8)

The reality ofdet4for realeandC-invariance require that the eigenvaluesenappear in the complex plane as quartets

en, en, or as imaginary pairs when m0. As ex-pected, the expansion of ln det4 in powers of ebegins in fourth order.

We have established that det4je¼0¼1 and that det4 has no zeros for real values of e. Therefore, by (2.6) detQED3>0for all reale, thereby allowing one to go from

(1.5) and (1.6). It might be objected that this is obvious, but we will need the detailed information introduced about det4in the sequel.

The determinantdet4 is an entire function ofe consid-ered as a complex variable, meaning that it is holomorphic in the entire complex e plane. Since P1_n¼1jenj3<1 for >0, its order is at most 3 [16,18]. This means that for any complex value of e, and positive constants A, K,

jdet4j< AðÞexpðKðÞjej3þÞ for any >0 From (1.6) and (2.6), and for real values ofe

ln det4 e2

4

Z _d3_k

ð2Þ3j

^

BðkÞj2

Z1

0

dz zð1zÞ

½zð1zÞk2þm21=2: (2.9)

This is a truly remarkable inequality. Referring to (2.9), det4’s growth is slower on the realeaxis than its potential growth in other directions. We also note thatdet4is largely unknown. Even the reduction of the fourth-order term in its expansion to an explicitly gauge invariant form involving onlyBfields requires a huge effort when the fields are not constant [19]. The sixth-order reduction has not been com-pleted as far as the author knows.

C. Upper bound ondetren

Insert (2.6) in (2.5) and get

ln det_ren¼e

2_T

42

Z _d3_k

ð2Þ3jB^ðkÞj 2Z1

0

dzzð1zÞ

ln

zð1zÞk2þm2 m2

þT

Z1

m2

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p ln det₄ðM2_{Þ: (2.10)}

The objective here is to obtain the behavior of lndetren when the coupling eis large, real, and positive. Since e always multiplies B, we introduce the scale parameter

is finite will be explained in Sec.III B. Then the integral in (2.10) is broken up intoReB_m2 and

R₁

eB.

Substitution of (2.9) into the lower range integral gives

ln detren e2_T

42

Z d3_k

ð2Þ3jB^ðkÞj 2Z1

0

dzzð1zÞ

ln

_{4eBþ2zð1zÞk}2_2m2

m2

þT

Z1

eB

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p ln det₄ðeB; M2_{Þ: (2.11)}

We have simplified the argument of the logarithm using 2pffiffiffiffiffixyxþyforx,y0. Then foreB m2,

lndet_rene

2_TjjBk2

242 ln

4eB m2

þT

Z1

eB dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p lndet4ðeB;M2Þ

þO

_eTR_d3_x Br2B

B

: (2.12)

The integral in (2.12) can be estimated by making a large mass expansion of ln det4. This is facilitated by inserting (2.6) in (2.4) and examining the smalltregion ofln det4’s resulting proper time representation. The details of this expansion are in Sec. 3B of [10], and give the result

ln det₄ðeB; M2_{Þ ¼} M!1

1 2

Z1

0 dt

t ð4tÞ

3=2_etM2Z

d3_x

2 45e

4_t4_ðBBÞ2_þOðe4_t5_{BBB r}2_BÞ

¼e4

R

ðBBÞ2

480M5 þO

e4RBBB r2B M7

: (2.13)

In the first line of (2.13), it is assumed that the heat kernel expansion is an asymptotic expansion intin the strict sense of its definition, namely [20]

< xjet½ðPeAÞ2eBjx >ð4tÞ3=2 X N

n¼0 anðxÞtn

t!0þð4tÞ3=2aNþ1ðxÞtNþ1: (2.14)

This must hold for every N. A necessary condition for (2.14) is thatB be infinitely differentiable to ensure that each coefficientanis finite. As far as the author knows, it is not known yet if this is a sufficient condition. So (2.14) is an assumption that may require additional conditions onB. Only coefficients an of Oðe2nÞ, n2 are present in ln det4’s expansion.

Thetintegration in (2.13), although extending to infin-ity, is limited to smalltsinceM! 1due to the parameter eBin (2.12). Substituting (2.13) in (2.12) results in

ln detren

e2_jjBk2_T

242 ln

_4eB

m2

þO

_e2_TR_ðBBÞ2

B2

þO

_eTR

B r2_B

B

; (2.15)

or

lim e!1

ln det_ren

e2_lne

kBk2_T

242 ; (2.16)

consistent with (1.4). This bound is independent of the charge renormalization subtraction point. If the subtraction

were made at photon momentum k2_¼2 _{instead of} k2 _{¼0, then the lnm}2 _{terms in (2.10) and (2.11) would} be replaced withln½zð1zÞ2_þm2_{, which has nothing} to do with strong coupling.

The scaling procedure used here is designed to obtain the least upper bound on ln detren. In [10], we chose to break up the M integral as Re_m42kBk4 and

R1

e4_kBk4. This

resulted in a fast1=e4_{falloff of theln det}

4terms compared to e2 _{here, but gave a weaker upper bound on ln det}

ren, namely

lim e!1

ln detren e2lne

kBk2_T

62 : (2.17)

We mention that the coefficient 1=960 in (3.16) in [10] should be1=360.

Here we might have chosen a more general scaling, such as eðlneÞB or eðln lneÞB, etc., with 1, >0. Then the right-hand side of (2.16) would have been re-placed withk_Bk2_T=242_{. The case <1causes the first} remainder term in (2.15) to be no longer subdominant. Therefore, our scalingeBis an optimal one.

III. LOWER BOUND ONdetren

A. Fundamentals

On referring to (2.5) the lower bound ondetrenwill come from operations onln detQED3. We begin with the operator

ln det_QED3 ¼ 1 2

Z1

0 dt

t Tr

eZt 0

dseðtsÞðPeAÞ2

BesðPeAÞ2

þe2Zt 0

ds₁Zts1 0

ds₂eðts1s2Þ½ðPeAÞ2eB

Bes2ðPeAÞ2_Bes1ðPeAÞ2

em2_t þ1

2

Z1

0 dt

t Trðe P2_t

eðPeAÞ2_t Þetm2

: (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 of scalar QED₃,

ln detSQED3 ¼

Z1

0 dt

t e tm2

TrðeP2teðPeAÞ2tÞ: (3.2)

Thus,

lndetQED3¼lndetSQED3

e2

2

Z1

0 dt

t e tm2

TrZt

0 ds1

Zts1

0

ds2eðts1s2Þ½ðPeAÞ

2_eB

Bes2ðPeAÞ2_Bes1ðPeAÞ2

;

(3.3)

remembering that the factor1=2in the last term of (3.1) is canceled by the spin trace. LetA¼ ½ðPeAÞ2þm21. In AppendixB, it is shown that1A=2B

1=2

A 2I2; that is, it is a Hilbert-Schmidt operator providedB2L2_{andm0. Then (2.7) gives}

ln det₂ð1e1_A=2_B1_A=2Þ ¼ln det½ð1e_A1=2_B1_A=2Þee_A1=2B1_A=2

¼Tr ln½ð1e1_A=2B_A1=2Þee_A1=2B1_A=2

¼TrZ

1

0 dt

t e tm2

ðeðPeAÞ2_t

e½ðPeAÞ2_eBt

þeAB

¼ e2Z1 0

dt t e

tm2Zt

0 ds1

Zts1

0

ds2Trðeðts1s2Þ½ðPeAÞ

2_eB

_Bes2ðPeAÞ2_Bes1ðPeAÞ2Þ_{: (3.4)}

In going from the penultimate to the last line in (3.4), use was again made of the identity (A2). Substituting (3.4) in (3.3) gives

ln detQED3 ¼ 1

2 ln det2ð1e

1=2

A B

1=2 A Þ

þln detSQED3: (3.5)

Asln detQED3andln det2 are well-defined by our choice of

fields, so is ln detSQED3 in (3.5). What has been

accom-plished here is to isolate the Zeeman termBinln det2. Since 1_A=2B1_A=2 is Hilbert-Schmidt and self-adjoint, ln det2 is susceptible to extensive analytic analysis.

Substitute (3.5) in (2.5):

lndetren¼ T

Z1

m2

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p 1

2lndet2ð1e

1=2 A B

1=2 A Þ

þlndetSQED3þ

e2_kBk2

24 ffiffiffiffiffiffiffiM2

p : (3.6)

We now introduce two central inequalities. The first relies on the diamagnetism of charged scalar bosons as expressed by Kato’s inequality in the form [21,22]

TrðeðPeAÞ2_t

Þ TreP2_t

: (3.7)

This implies that on average the energy eigenvalues of such bosons rise in a magnetic field and hence by (3.2) that [22]

ln det_SQED30: (3.8)

The second inequality is introduced beginning with the penultimate line of (3.4). Noting that the spin trace of the _ABterm is zero, then

lndet2ð1e1A=2B 1=2 A Þ

¼Z1

0 dt

t e tm2

TrðetðPeAÞ2

e½ðPeAÞ2_eBt

Þ: (3.9)

By the Bogoliubov-Peierls inequality [23,24]and Sec. 2.1, 8 of [25]

Tre½ðPeAÞ2_eBt

TretðPeAÞ2_t

etehBi_{; (3.10)}

where

hBi ¼TrðBe

tðPeAÞ2_t Þ

Hence,

ln det2ð1e1A=2B 1=2

A Þ 0; (3.12) which is consistent with (3.5) when combined with (1.6) and (3.8). There is another reason why (3.12) holds. Let C¼e1_A=2B1_A=2 SinceCis Hilbert- Schmidt,

ln det2ð1CÞ ¼ln det½ð1CÞeC

¼Tr½lnð1CÞ þC

¼1

2Tr lnð1C

2_Þ

¼1 2

X1

n¼1

lnð1 2

nÞ: (3.13)

The third line of (3.13) follows from the second since the trace over spin eliminates all odd powers ofC. In the last

line, we introduced the real eigenvalues _n of e1_A=2

B1_A=2. Sinceln det2is real and finite, thenj nj<1for all n, giving (3.12). Because1_A=2B1_A=2I₂, it is a com-pact operator, and so the n are countable and of finite multiplicity.

Now consider

@

@m2lndet2ðm 2_Þ

¼Z1

0

dtetm2

Trðe½PeA2_eBt

eðPeAÞ2_t Þ 0;

(3.14)

by (3.9), (3.10), and (3.11). Therefore,det2is a monotoni-cally increasing function of m2_{. Next, break up the M} integral in (3.6) as in Sec.II C:

ln detren¼ T

ZeB

m2

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2m2

p 1

2 ln det2ð1e

1=2

A B

1=2

A Þ þln detSQED3þ

e2_k Bk2

24 ffiffiffiffiffiffiffiM2

p

þT

Z1

eB

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p ln detQED3þ

e2_kBk2

24pffiffiffiffiffiffiffiM2

; (3.15)

where we reinserted (3.5) into the upper-rangeMintegral. By (3.14)

ZeB

m2

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p ln det₂ðM2_{Þ ln det} 2jM2_¼m2

ZeB

m2

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p ¼2 ln det₂ð1e1_A=2B1_A=2Þj_M2_¼m2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

eBm2

p

:

(3.16)

Hence, (3.8) and (3.16) result in (3.15) becoming

ln detren T

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

eBm2

p

ln det2ð1e1A=2B 1=2

A ÞjM2_¼m2þ

e2_TkBk2

242 ln

_eB

m2

þ e2T 122kBk

2_ln

1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1m

2

eB

s

þT

Z1

eB

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p ln det_QED3þ e

2_k Bk2

24pffiffiffiffiffiffiffiM2

: (3.17)

We now turn to the strong coupling behavior ofln det2.

B. Strong coupling behavior ofln det2

The eigenvalues _n in (3.13) are obtained from

e1_A=2B_A1=2’n¼ n’n; (3.18)

for ’_n2L2 _{following the remark under (B3) in} AppendixB. Letting1_A=2’_n¼ c_ngives

ðPeAÞ2eB

n

cn¼ m2cn; (3.19)

where cn 2L2 providedm0. This follows from (B5) and Young’s inequality (B7). The requirement thatm0 follows from the role of the eigenvaluesf _ng1_n¼1 as adjust-able coupling constants whose discrete values result in bound states with energy m2 for a fixed value of e.

Since the operator ðPeAÞ2eB0, such bound states are impossible unlessj nj<1for alln, which is the physical reason why (3.12) is true. Inspection of (3.19) suggests that as e increases j _nj likewise increases for fixed nto maintain the bound state energy at m2_{. This} is illustrated by the constant field case that is excluded from our analysis:

j nj ¼

jeBj

ð2nþ1ÞjeBj þm2; n¼0;1;. . .: (3.20)

c;

ðPeAÞ2e_B

c

c;

ðPeAÞ2

e

jBj

c: (3.21)

Thus, the Hamiltonian on the left,Hþ, dominates that on the right,H. Let Nm2ðHÞ denote the dimension of the

spectral projection onto the eigenstates of HamiltonianH with eigenvalues less than or equal to m2_{. Because} HþH, then Nm2ðHþÞ Nm2ðHÞ. N_m2ðHþÞ is

an overestimate of the number of the bound states ofHþ

at m2 for a fixed value of but satisfactory for our purpose here.

By the Cwinkel-Lieb-Rozenblum bound in the form [26]

Nm2ðHÞ C

Z

d3x

e

jBðxÞj m2 ₃₌₂

þ ; (3.22)

where½aþ¼maxða;0ÞandC¼20:1156. The factor 2 accounts for the additional spin degrees of freedom in the present estimate. Since j nj ¼Oð1Þ, we are confident that the degeneracy/multiplicity associated with each _n in (3.13) does not exceed cjej3=2R_d3_xjBj3=2_{, where c}

0:2312 is another finite constant. This estimate has to be modified for values ofn > Nbeyond which _nassumes its asymptotic form as discussed below. Therefore, fornN we will estimate the sum in (3.13) by factoring out the common maximal degeneracycjej3=2Rd3xjBj3=2and treat each nin the factored sum as having multiplicity equal to one. Those n, if any, that vanish as e! 1 give a sub-dominant contribution toln det2 in (3.13) since by inspec-tion their contribuinspec-tion grows at most as 2

nje= nj3=2. We now turn to the largeedependence of _n. From here on we assume that cn is normalized to one. By C-invariance we may assume e >0. Now consider the expectation value of (3.19):

hnjðPeAÞ2_jni e n

hnjBjni ¼ m2_{: (3.23)}

From (3.23) ifhnjBjni>0then n>0andvice versa. Therefore, we need only consider n>0and write

n¼

_hnjð

Pe_AÞ2_jni ehnjBjni þ

m2 ehnjBjni

₁

; (3.24)

wherehnjBjni0as (3.23) must be satisfied. The case n ¼0 for some n corresponding to hnjBjni ¼0 can be ignored as _n¼0contributes nothing toln det₂ in (3.13). An easy estimate gives

jð_cn;BcnÞj ðcn;jBjcnÞ max

x jBðxÞj: (3.25) Because B2L2 _{and is assumed infinitely differentiable,} then maxxjBj is finite. Hence, hnjBjni is a bounded function ofeandn.

Now consider the ratio Rn¼ hnjðPeAÞ2jni= ehnjBjni in (3.24). The case Rn!e 10 is ruled out

since this implies _n! 1. The case R_n!_e 11 implies n!0, which gives a subdominant contribution to (3.13) as discussed above. The final possibility is 1R_n<1 for e! 1. The case R_n!1 for e! 1 happens if hnjðPeAÞ2_{jni ehnjBjni. Since c}

n2L2,

hnjðPeAÞ2_{eBjni ¼0 implies ðPeAÞ}c n¼ 0. Now this may happen for the B fields considered so far. But if we exclude zero-mode supportingBfields [27] from our analysis it cannot. By so doing, we can exclude the case n¼1nðeÞ,nð1Þ ¼0. We will see below why this is necessary.

We proceed to estimate the strong coupling limit of ln det2 in (3.13). First, consider the sum for nN. We need only consider0<j nj<1for alle, includinge¼ 1 as concluded above. Hence, on factoring out the common maximal multiplicity of the _nwe get

lim e 1

XN

n¼1

lnð1 2 nÞ

c1e3=2

Z

d3_xjBj3=2_{; (3.26)}

wherec1is a constant and noting again that the eigenvalues n!0are subdominant.

Since n!0forn! 1and1=2 jlnð1 2nÞ= 2nj 3=2for 2n<1=2, the absolute convergence of the series in (3.13) requiresP1_n¼1 2

n<1. Consider this sum forn > N and indicate the degeneracy factorsn explicitly:

S X 1

n>N

nðeÞ 2nðeÞ: (3.27)

We estimated from (3.22) that ncje= nj3=2

R

d3_xjBj3=2_{. So}

Sce3=2Z d3xjBj3=2 X 1

n>N

no degeneracy

j _nj1=2_{<1: (3.28)}

This implies that forn > N

j _nðeÞj ¼CnðeÞ

n2þ ; (3.29)

where >0andC_nis a bounded function ofnandewith lim_e!1CnðeÞ<1. Otherwise, j nj<1for any ncannot be satisfied. Accordingly, the series in (3.28) is uniformly convergent ineby the WeierstrassMtest and so

lim e!1

X1

n>N

lnð1 2 nÞ

=e3=2 c2

Z

d3_xjBj3=2_{; (3.30)}

wherec2is a constant. From (3.13), (3.26), and (3.30), we conclude

lim

e!1jln det2ð1e 1=2

A B

1=2 A Þj=e3=2

c₃Z d3_xj

Bj3=2; (3.31)

As a check on (3.31), refer to (3.5). ForB2L3=2_ðR2_Þ, we found [10]

ln det_QED3 Ze

3=2

6

Z

d2_xjBðxÞj3=2_{; (3.32)}

forBðxÞ 0orBðxÞ 0,x2R2.Z is the dimension of the remaining space box. We know that ln det20 and

ln detSQED30 in (3.5). Specializing (3.31) to these B

fields, it is seen that the strong coupling growth ofln det2 is consistent with (3.32).

Finally, if zero-mode supportingBfields were allowed, we would have obtained ln det₂¼_e 1Oðe3=2lnðeÞÞ, ðeÞ!e 10, since when j nje 11nðeÞ, nð1Þ ¼0, the logarithm in (3.13) gives an additional factor lnn. As will be seen below the limit (1.4) requires lim_e!1ln det2=e3=2 ¼finite (or zero).

C. Strong coupling limit of (3.17)

It remains to estimate the large coupling limit of the last term in (3.17),

IT

Z1

eB

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2m2

p ln detQED3þ

e2_kBk2

24 ffiffiffiffiffiffiffiM2

p

¼T

Z1

eB

dM2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

M2_m2

p e2 4

Z _d3_k

ð2Þ2jB^ðkÞj 2

Z1

0

dz zð1zÞ

½zð1zÞk2þM21=2þ

e2_kBk2

24pffiffiffiffiffiffiffiM2

þln det4ð1eS6AÞ

; (3.33)

where we substituted (2.6) forln detQED3. Calculation of the

first two terms in (3.33) is straightforward. The last term has already been estimated in Sec.II and is given by the second term in (2.15). Hence,

I¼O

_e2_TR_ðBBÞ2

B2

þO

_eTR

B r2_B

B

: (3.34)

Taking into account (3.31) and (3.34), we obtain from (3.17)

lim e!1

ln det_ren

e2_lne

kBk2_T

242 : (3.35)

Equations. (2.16) and (3.35) therefore establish (1.4).

IV. SUMMARY

The two assumptions underlying (1.4) are first that the continuum limit of the lattice diamagnetic inequality co-incides with (1.5), and second that the heat kernel expan-sion of the Pauli operator in (2.14) is an asymptotic series. These assumptions can and should be proven or falsified.

In addition, the result (1.4) assumes that the vector potential and magnetic field satisfy the following conditions:

B2L2_ðR3_{Þto defineln det}

renin (2.5) and to ensure that

_A1=2B_A1=22I₂ following Appendix B. In addition B2L3=2_ðR3_{Þ in order that the degeneracy estimate in} (3.22) is defined. To ensure that the bound in (3.31) holds, zero-mode supportingBfields are excluded. Also,Bmust be infinitely differentiableðC1Þ to ensure that the expan-sion coefficients in (2.14) are finite.

If A is assumed to be in the Coulomb gauge, then by (2.3) _A2L6_ðR3_{Þ. If}

B2L3=2_ðR3_{Þ, then}

A2L3_ðR3_Þby the Sobolev-Talenti-Aubin inequality [12]. In order to define det_QED3, it is necessary to assume A2Lr_ðR3_Þ, r >3, following the discussion under (2.6). IfA2L3ðR3Þ andL6ðR3Þ, thenA2LrðR3Þ,3< r <6also. This follows from Ho¨lder’s inequality [28]

kfgkr jjfkpkgjjq; (4.1)

withp1þq1 ¼r1,p,q,r1. SinceB¼rAand B2C1, thenA2C1.

We note that the sample functions AðxÞ supporting the Gaussian measure in (1.2) with probability one are not C1. It is generally accepted that they belong to

S0_ðR4_{Þ, the space of tempered distributions. Therefore,} we point out here that the C1 functions we introduced can be related to A 2S0ðR4Þ by the convoluted field A

ðxÞ¼

R

d4_yf

ðxyÞAðyÞ2C1, providedf2SðR4Þ, the functions of rapid decrease. Then the Fourier transform of the covarianceRdðAÞA

ðxÞAðyÞderived from (1.2) is ^

DðkÞjf^ðkÞj2, where f^ 2C1. Since QED4 must be ultraviolet regulated before renormalizing, f^ can serve as the regulator by choosing, for example, f^ ¼1, k2 2, andf^ ¼0,k222. So the need to regulate can serve as a natural way to introduce C1 background fieldsA

intodetren—but not the rest ofdin (1.1)—and into whatever else one is calculating. This procedure is a generalization of that used in the two-dimensional Yukawa model [29].

Finally, the obvious generalization of (1.4) for an ad-missible class of fields onR4 is

lim e!1

ln det_ren

e2_lne ¼

1 482

Z

d4_xF2

ðxÞ: (4.2)

There is no chiral anomaly term since F falls off faster than 1=jxj2 _and R_d4_xF~

F¼

R

d4_x@

ðAFÞ ¼0., where F~¼12F. Equation (4.2) remains to be verified.

If (1.4) and (4.2) do indeed indicate instability, then they are yet another reason why QED should not be considered in isolation.

APPENDIX A

Ft¼etðXþYÞetX:

Then

dF_t dt ¼ e

tðXþYÞ_YetX_:

Integrating gives

etðXþYÞ_etX_¼ Zt 0

dseðtsÞðXþYÞ_YesX_{; (A1)}

known as Duhamel’s formula. Iterating once gives the required identity:

etðXþYÞetX

¼ Zt

0

dseðtsÞX_YesX_þZt 0

ds1

Zts1

0

ds2eðts1s2ÞðXþYÞYes2XYes1X: (A2)

APPENDIX B

Here we show that the operatorK¼1_A=2B1_A=22I2 and hence that K is Hilbert-Schmidt. This follows [14,15,28] if and only if K is a bounded operator on L2_ðR3_{; d}3_x;C2_{Þhaving a representation of the form}

ðKfÞðxÞ ¼Z Kðx; yÞfðyÞd3_{y; f2L}2_{; (B1)}

where

Kðx; yÞ ¼ hxj1_A=2B1_A=2jyi; (B2)

and whereK2L2_ðR3_R3_;d3_xd3_{yÞ. Moreover,}

kKk2 2 ¼

Z

jKðx; yÞj2_d3_xd3_{y: (B3)}

If it can be shown that K2L2, then it trivially follows thatKmapsL2 into itself. So consider

kKk_L2 ¼2

X

i

Z

d3_xd3_yB

iðxÞAðx; yÞBiðyÞAðy; xÞ

2X i

Z

d3_xd3_yjB

iðxÞkAðx; yÞkBiðyÞkAðy; xÞj:

(B4)

A form of Kato’s inequality [5,21,31] asserts that the interacting scalar propagator is bounded by the free propa-gator

j_Aðx; yÞj ðxyÞ; (B5)

whereðxÞ ¼ ð4jxjÞ1emjxj _{in three dimensions. Then}

kKk_L2 1 82

X

i

Z

d3xd3yjBiðxÞ

1 jxyj2e

2mjxyj

BiðyÞj: (B6)

By Young’s inequality in the form [23]

Z

d3_xd3_{yfðxÞgðxyÞhðyÞ}

kfkpkgkqjjhkr; (B7)

where p1þq1þr1¼2, p, q, r1, and kfk_p¼

ðRd3_xjfðxÞjp_Þ1=p _{etc., obtain from (B6) with p¼r¼2,} q¼1

kKk_L2 1 82

X

i

jjB_ik2Z _d3_xe2mjxj_=x2

¼ kBk2_ð4mÞ1_{: (B8)}

Therefore, by the theorem that began this appendix, 1_A=2B1_A=22I2 whenm0andB2L2. We men-tion that this can be proved even when m¼0 provided B2L3=2.

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