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arXiv:2110.13775v1 [math.SP] 26 Oct 2021

Horizontal magnetic fields and improved

Hardy inequalities in the Heisenberg group

Biagio Cassano

?Valentina Franceschi†David Krejcir´ık‡

Dario Prandi

In this paper we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of thecorresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov-Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland-Stein operator, that we prove in this paper and has an interest on its own.

1. Introduction

It is well known that the (elliptic) Laplacian inRnwithn≥3 satisfies the Hardy inequality (in the sense of quadratic forms for self-adjoint realizations of the respective operators inL2(Rn)) -Δ≥cn ?2inL2(Rn),(1.1) where?(x) :=? x21+···+x2nis the Euclidean distance to the origin ofRnandcn:= (n-2)2/4 is a dimensional constant. Moreover, the inequality is optimal in the sense that no positive term can be added to the right-hand-side of (1.1), see [53, Sec. 8.1]. On the other hand, ifn= 1 orn= 2, there exists no positive functionwsuch that-Δ≥w. These properties are termed as thecriticalityversus thesubcriticalityof the Laplacian in ?Department of Mathematics, Universit`a degli Studi di Bari"A. Moro", via Orabona 4, 70125, Bari,

Italy;biagio.cassano@uniba.it.

†Dipartimento di Matematica Tullio Levi-Civita, Universit`a di Padova; valentina.franceschi@unipd.it.

‡Department of Mathematics, Faculty of Nuclear Sciences andPhysical Engineering, Czech Technical

University in Prague, Trojanova 13, 12000 Prague 2, Czechia;david.krejcirik@fjfi.cvut.cz.

§Universit´e Paris-Saclay, CNRS, CentraleSup´elec, Laboratoire des signaux et syst`emes, 91190, Gif-sur-

Yvette, France.dario.prandi@centralesupelec.fr.

1 the low and high dimensions of the Euclidean space, respectively. These notions naturally extend to the setting of more general elliptic operators andRiemannian manifolds, where they coincide with the alternative concepts of theparabolicity/recurrencyversus thenon- parabolicity/transiency(see [47] for an overview). In 1998 Laptev and Weidl demonstrated in [39] that adding a magnetic field to the Laplacian inR2makes the operator subcritical, meaning that there is a Hardy-type inequality. The result has stimulated an enormous growth ofinterest in magnetically induced Hardy-type inequalities for elliptic operators with many important applications in quantum mechanics and elsewhere. More generally (see [52, 9]), while the shifted operator-Δ-cn/?2is critical inL2(Rn) for alln≥2, it becomes subcritical after adding a magnetic field to the Laplacian. The purpose of the present paper is to study the influence of magnetic fields on the criticality properties of the (sub-elliptic) Laplacian inthe Heisenberg groupH1. The latter is the foremost example of sub-Riemannian structureonR3formally defined by the completely non-integrable distributionD:= span{X,Y}, where

X:=∂x-y

2∂z, Y:=∂y+x2∂z,(1.2)

with (x,y,z)?R3. The directions ofDare calledhorizontal directions, and fixing onD the metric for which{X,Y}is an orthonormal frame allows to define a distance on H

1. The associated Laplacian is the sub-elliptic operator-Δ :=-X2-Y2, that has

been extensively studied in the last fifty years due to its deep connections with diverse subjects; see, e.g., [33, 21, 22, 34, 43]. Similarly to (1.1), it is known due to Garofalo and Lanconelli [27] that the optimal

Hardy-type inequality

-Δ≥r2

ρ4inL2(H1) (1.3)

holds, whereρ(x,y,z) :=4? (x2+y2)2+ 16z2is the Koranyi norm andr(x,y,z) :=? x2+y2is the radial distance to thez-axis. Notice thatr2/ρ4=|?ρ|2/ρ2, where |?ρ|2=|Xρ|2+|Y ρ|2is the norm of thehorizontal gradientofρ, associated with the sub-Riemannian structure ofH1. Further results on Hardy-type inequalities on the Heisenberg group are presented in [45, 13, 23]. We refer to [28, 29, 51, 50, 11, 10, 41] for extensions to more general Carnot Groups, and to [1, 12, 14, 15, 24, 36] for extensions to sub-elliptic operators that are not necessarily inducedby groups. One of the main results of the present paper states that, while the shifted operator -Δ-r2/ρ4is critical inL2(H1), it becomes subcritical after adding a magnetic field to the Laplacian. In this way we establish a sub-Riemannian analogue of the celebrated result of Laptev and Weidl [39]. Other functional inequalities in the Heisenberg group and their respective magnetic improvements are also investigated. 2

2. Main results2.1. Horizontal magnetic fields on the Heisenberg groupTo state our main results, we first need to properly introducemagnetic fields in the

Heisenberg group. The notion of magnetic fields and associated magnetic operators are naturally formulated in terms of differential forms. On a Riemannian manifoldM, amagnetic fieldBis a closed 2-form, i.e.,dB= 0 wheredis the exterior differential. The correspondingmagnetic potentialAis a 1-form such thatdA=B. The latter allows one to define the classical and quantum dynamics under the influence of the magnetic field as follows. Classically, given a Hamiltonian h?C∞(T?M), the magnetic Hamiltonian ishA(p,q) :=h(p+A(q),q), whereqandp are the (generalized) coordinates and momenta onM; the magnetic classical trajecto- ries are defined via standard Hamiltonian equations. The magnetic quantum dynamics are then obtained by the Schr¨odinger equation with respectto an appropriate quantiza- tionHAofhA. In the case ofhbeing the free Hamiltonian onM, the quantum magnetic

HamiltonianHAcoincides with themagnetic Laplacian

-ΔA:= (-i?+A)2,(2.1) where?is the Riemannian gradient; or equivalently, ΔA=?A2, where?A:=?+iA is themagnetic gradient. Of course,-Δ0=-Δ. In the context of the Heisenberg groupH1, it is natural to define the magnetic fields in such a way that the corresponding classical magnetic trajectories retain the property of beinghorizontal. As we will show later in Section 4.2, this is not really a requirement inasmuch it is a natural consequence of the definition of classical magnetic trajectories. This naturally leads to define magnetic fields in the context of theRumin complex[49]. The latter roughly corresponds to consider vector potentials modulo the contact formω definingD(i.e.,A=Axdx+Aydymodω) and magnetic fields as horizontal 2-forms (i.e., such thatB?ω= 0). Then, formally as above, the magnetic (sub-)LaplacianinH1 is given by (2.1) with the only difference that?u:= (Xu)X+ (Y u)Yis the now the horizontal gradient. Let us remark that the Rumin complex has already been applied to derive Maxwell"s Equations in the more general setting ofCarnot groups in [26, 25]. To the best of our knowledge, the notion of the Heisenberg sub-Laplacian with a magnetic field has not been studied yet, except for [54]. We discuss this paper in Ap- pendix A, where we raise a crucial criticism of its main result. We also mention [3] where the authors investigate improved Hardy inequalities in theGrushin setting in presence of Aharonov-Bohm magnetic fields. However, the magnetic operator introduced in that work, when interpreted in our setting, corresponds to a magnetic sub-Laplacian with an extra electric potential.

2.2. Uniform magnetic fields

In a Riemannian manifoldM, it is well known that the magnetic field has a deep influence on spectral properties of the magnetic Laplacian-ΔA(when realized as a self-adjoint 3 operator inL2(M)). Roughly, the magnetic field acts as a repulsive interaction, which is known as thediamagnetic effectin quantum mechanics. The diamagnetic effect is best seen in the case of uniform (or homogeneous) fields in the planeR2, i.e.,B(x,y) =bdx?dywithb?R. In this case, one has (see [42, Corol. 2.5] and [43, Thm. 3] for higher dimensions and a Riemannian counterpart, respectively) infσ(-ΔA) =|B|inL2(R2),(2.2) whereAis any 1-form inR2such thatB=dAand we write|B|:=|b|. Sinceσ(-Δ) = [0,∞), it is clear that any non-trivial uniform magnetic field uplifts the bottom of the spectrum. This fact particularly implies that the heat semigroup associated with-ΔAin L

2(R2) admits a faster decay rate once the magnetic field is turned on. As an immediate

consequence of (2.2), one has the optimal Poincar´e-type inequality-ΔA≥ |B|in the sense of quadratic forms inL2(R2). This inequality extends to the case of variable magnetic fields of strength bounded from below by the positive constant|b|. Our first result is the following generalization of (2.2) to the Heisenberg group. Theorem 2.1.LetB(x,y,z) =b1dx?ω+b2dy?ωwithb:= (b1,b2)?R2. Then infσ(-ΔA) =c|B|2/3inL2(H1),(2.3) whereAis any 1-form inH1such thatB=dA,|B|:=? b21+b22, andc >0is a universal constant. The non-linear growth in the strength|b|of the magnetic field in (2.3) is ultimately related with the results obtained in [43], see Remark 5.1 below. As above, (2.3) implies the optimal Poincar´e-type inequality-ΔA≥c|B|2/3in the sense of quadratic forms inL2(H1). This is again a non-trivial diamagnetic improvement due to the magnetic field, becauseσ(-Δ) = [0,∞) in the Heisenberg case as well. Note that the results (2.2) and (2.3) depend onBonly, while they are independent ofA. This is natural and physically expected because of the vanishing of the first cohomology group ofRn. More specifically, given any 0-formf, the operators-ΔAand -ΔA+dfare unitarily equivalent, so isospectral. This is known asgauge invarianceof the magnetic field in quantum mechanics.

2.3. Aharonov-Bohm magnetic potentials

Interesting and unexpected phenomena appear for more complex geometries when the gauge invariance does not hold. In this setting, the non-exact vector potentialsAyielding null magnetic fields (i.e., such thatdA= 0 butA?=dffor any smooth functionf) are known asAharonov-Bohm potentials. The simplest example is the punctured planeR2\ {0}with the magnetic potential A α:=αd?whereα?Rand (?,?)?(0,∞)×S1are polar coordinates. It is important to notice thatAαis actually a strongly singular vector potential; indeed, in Cartesian coordinates one hasAα=α?(x,y)-2(xdy-ydx). In particular,Aαis not locally square integrable inR2. 4 Although classically invisible (indeed,dAα= 0 inR2\{0}), the Aharonov-Bohm po- tentialAαstill has a strong influence on spectral properties of-ΔAα, and therefore on quantum dynamics. Indeed, it can break the essential self-adjointness of the magnetic Laplacian [2, 16]. Moreover, even in the case of the Friedrichs extension, where the spectrum stays unchanged, it was observed by Laptev and Weidl in [39] that the pres- ence of the Aharonov-Bohm potential makes the magnetic Laplacian subcritical. More specifically, one has the optimal magnetic Hardy inequality -ΔAα≥d(α,Z)2 ?2inL2(R2\ {0}).(2.4) The case of integer flux quanta, i.e.α?Z, must be excluded, because-ΔAαwith any suchαis unitarily equivalent to-Δ, which is critical. Our next result is the following generalization of (2.2) to the Heisenberg group. In this case, it is natural to work in cylindrical coordinates (r,?,z)?(0,∞)×S1×R. Theorem 2.2.LetAbe an Aharonov-Bohm potential onH1\ ZwithZ:={(0,0,z) : z?R}. Then, up to gauge invariance,A=Aα:=αd?modωfor someα?Rand the inequality -ΔAα≥d(α,Z)2 r2inL2(H1\ Z) (2.5) holds in the sense of quadratic forms, wherer(x,y,z) :=? x2+y2. Moreover, the inequality is optimal in the sense that no positive functioncan be added to the right- hand side of(2.5). This is indeed a non-trivial magnetic improvement because (even if-Δ inL2(H1) is subcritical, see (1.3)) there exists no positive numbercsuch that-Δ≥c/r2in L

2(H1\ Z), see [51]. On top of this, Theorem 2.2 will be instrumental to the proof of

Theorem 2.4, where we show that the critical operator-Δ-r2/ρ4becomes subcritical after adding a magnetic field to the Laplacian. The proof of Theorem 2.2, presented in Section 6.1, requiresa careful analysis of the commutation relations between the Aharonov-Bohm magnetic potential and both horizontal vector fields. This is in contrast with the relatively simple proof of (2.4) based on the polar decomposition of the Euclidean gradient into a "radial" and an "angular" direction, which does not exist in the Heisenberg group (see[23], and Remark 6.1 below). In the Euclidean case, an analogous decomposition in hypercylindrical coordinates (r,?,z)?(0,∞)×S1×Rn-2allows for even stronger improvements. Namely, in [20], the authors consider Aharonov-Bohm potentialsAα:=αd?inRn\ {r= 0}, where n≥2 andr(x) :=? x21+x22is the distance to the subspace{r= 0}of dimensionn-2, and prove the following inequality for the Euclidean magnetic Laplacian: -ΔAα-cn ?2≥d(α,Z)2r2inL2(Rn\ {r= 0}),(2.6) where?(x) :=? x21+···+x2nis the Euclidean distance as in (1.1). That is, the Aharonov-Bohm potential improves the classical Hardy inequality (1.1) with a weight 5 singular at the origin by a term that is singular on the subspace{r= 0}. Forn= 2, (2.6) reduces to (2.4). Motivated by this fact, in Section 7.1, we study how Aharonov-Bohm potentials of Theorem 2.2 interact with the Hardy-type inequality (1.3) due to Garofalo and Lan- conelli, the latter being classical in the case of the Heisenberg group. The main result of this section is Proposition 7.1 where we show that the following improvement holds under suitable symmetry assumptions: -ΔAα-r2 ρ4≥d(α,Z)21- |?ρ|4r2inL2(H1\ Z).(2.7) Here,ρis the Koranyi norm, and?ρits horizontal gradient. In particular, the above inequality holds for functions that are symmetric with respect to rotations around the z-axis, or with respect to the reflection (x,y,z)?→(x,y,-z). We stress that these sym- metry assumptions frequently appear in functional inequality concerning the Heisenberg group, see, e.g., [44]. The crucial observation allowing to derive the improvement(2.7) is the connection detailed in Lemma 7.1 below between the Aharonov-Bohm magnetic Laplacian ΔAα and the Folland-Stein operator [22] L

α:=-Δ-iα∂zinL2(H1\ Z).(2.8)

This allows us to deduce (2.7) from the following Hardy-typeinequality forLα, proved in Section 7.1, which is interesting in its own right.

Theorem 2.3.Letα?(-1,1). The inequality

L

α≥(1-α2)r2

ρ4inL2(H1\ Z) (2.9)

holds in the sense of quadratic forms. Moreover, the inequality is optimal in the sense that no positive function can be added to the right-hand sideof(2.3).

2.4. Mild magnetic fields

We now turn our attention to physically more relevant magnetic fields, which lie in between the extreme situations of uniform and Aharonov-Bohm fields. We call them mildfor they are at the same timeregular, in the sense that they are realized by smooth magnetic potentials (contrary to the Aharonov-Bohm potentials), andlocalin the sense that they vanish at infinity (contrary to the uniform fields).Thenσ(-ΔA) =σ(-Δ) and the quantification of the magnetic effects is more subtle. In the Euclidean case, it is known that-ΔA-cn/?2is subcritical inL2(Rn) ifn≥2 andBis not identically equal to zero (recall that-Δ-cn/?2is critical because of the optimality of (1.1)). In the general setting, this was first observed by Weidl in [52], who established the Hardy-type inequality-ΔA-cn/?2≥c(n,A,Ω)χΩwith any compact subset Ω?Rnandc(n,A,Ω) being a positive constant depending onn≥2,A?= 0 and Ω, 6 whereχΩdenoted the indicator function of Ω. The compactly supported Hardy weight on the right-hand side of this inequality can replaced by a positive one [9, Thm. 1.1]: -ΔA-cn ?2≥c(n,B)1 +r2log2rinL2(Rn),(2.10) valid for every smoothAsuch thatdA=B, wherec(n,B) is a positive constant de- pending onn≥2 andB?= 0. Under extra hypotheses, it is next possible to remove the logarithm from the right- hand side of (2.10) (see [9, Thm. 3.2] based on ideas of [39]).In particular, this is the case ifn= 2,B(x,y) =b(x,y)dx?dywith a smooth functionb:R2→Rand the total magnetic flux B:=1

2π?

R

2b(x,y)dxdy(2.11)

is not an integer. A key observation is that, by Stokes theorem, the vector potential of a compactly supported magnetic fieldBcan be chosen as the Aharonov-Bohm potential

Bd?outside a compact neighborhood of the origin.

In the present setting of the Heisenberg group, we are primarily concerned with im- proving the Hardy-Garofalo-Lanconelli inequality (1.3) due to the presence ofanymag- netic field. Theorem 2.4.LetΩ?H1be a bounded open set with Lipschitz boundary, andAbe a vector potential of either one of the following types: (i) a smooth vector potential onH1whose associated magnetic fieldB=dAis such thatB?≡0onΩ; (ii) an Aharonov-Bohm potential onH1\ Zwith non-integer flux (i.e., up to gauge invariance,A=αd?modωwithα?R\Z). Then, there exists a positive constantc(A,Ω)dependent onAandΩsuch that -ΔA-r2 Moreover, ifAis of type (i),c(A,Ω)depends only on the associated magnetic fieldB. This result is reminiscent of that of Weidl [52] in the Euclidean case mentioned above. We prove it in Section 7.2 by showing that the spectrumof the shifted oper- ator-ΔA-r2/ρ4with Neumann boundary conditions on the bounded set Ω is purely discrete and bounded away from 0 whenever the vector potentialAsatisfies the assump- tions of Theorem 2.4. We stress that the proof that we presentin the Aharonov-Bohm case relies on the validity of the improved Hardy inequalityfrom the center presented in Theorem 2.2. We leave as an open problem whether the compactly supported Hardy weight on the right-hand side of (2.12) can be replaced by a positive one in the spirit of (2.10). 7 Finally, we present Hardy-type inequalities, which do not necessarily improve (1.3), but providepositiveHardy weights under magnetic flux conditions. We restrict tomag- netic fields of the form B(x,y,z) =b1(x,y,z)dx?dω+b2(x,y,z)dy?ω(2.13) in the sense that it is the case of the functionsb1,b2:R3→R. Notice, in particular, thatBcould be unbounded with respect to the variablez. It is useful to remark that, due to its closedness,Bis uniquely determined by itsprimitive, that we define as b(rcos?,rsin?,z) :=-? r b

1(tcos?,tsin?,z)dt,(2.14)

wherer≥0,??S1andz?R. Theorem 2.5.LetBbe a magnetic field onH1of the form(2.13)and assume that its F B:=1

2π?

R

2b(x,y,z)dxdy,(2.15)

is independent ofz?R. Moreover, ifFB/?Z, there exists a positive constantc(B) dependent onBsuch that -ΔA≥c(B)

1 +r2inL2(H1).(2.16)

Let us observe that ifBis compactly supported thenFB= 0. In this case, it can actually be shown that the associated magnetic potential can be chosen to be compactly supported, which implies that no improvement of-ΔA≥0 as above is possible, see

Remark 6.3 below.

Remark 2.1.In Theorem 6.1 below we prove a slightly stronger result thanTheo- rem 2.5. Indeed, we are able to replace the right-hand side of(2.16) with a function that behaves as (rlogr)-2asr↓0. We stress that the same technique can be applied also in the Euclidean case, yielding an improvement of (2.10), thatwas not known to the best of our knowledge, see Remark 6.6. The Euclidean magnetic Hardy-type inequalities (2.10) were fundamental ingredients for the study of the large-time behavior of the magnetic heatsemigroup in [37, 9]. In particular, it was shown that although compactly supported magnetic fields on the Euclidean plane do not shift the spectrum, if they have non-integer flux, they improve the decay of theL2-norm of the solutions of the heat equation with initial dataliving in some appropriate weighted space. An interesting research direction for a future work is the application of Theorem 2.4 to show an analogous improved decay rate of the magnetic heat semigroup in the Heisenberg setting. 8

3. The Heisenberg group3.1. The basic structureThe Heisenberg groupH1isR3endowed with the non-commutative group law

(x,y,z)?(x?,y?,z?) =? x+x?,y+y?,z+z?+xy?-x?y 2? where (x,y,z),(x?,y?,z?)?R3. A basis for the Lie algebra of left-invariant vector fields is given byX,Ydefined in (1.2) together withZ:=∂z. The associated sub- Riemannian structure is given by the distributionD:= span{X,Y}endowed with the scalar product makingXandYorthonormal, and denoted by a dot,i.e.,(a1X+a2Y)· (b1X+b2Y) =a1b1+a2b2forai,bi?C. This structure is step 2 since [X,Y] =Z, so that span{X,Y,[X,Y]}|q=TqH1for allq?H1. Moreover it is nilpotent since [X,Z] = [Y,Z] = 0. What is relevant for the following is that the Heisenberg group structure iscontact.

That is, the one formω?Ω1(H) given by

ω(x,y,z) :=dz-1

2(xdy-y dx)

satisfiesω?dω?= 0 and kerω= span{X,Y}. The Reeb vector field isZ(i.e.,Z? kerdωandω(Z) = 1), and the dual basis of the cotangent bundleT?H1associated with {X,Y,Z}is{dx,dy,ω}.

3.2. The Laplacian

The sub-Riemannian Hamiltonianh?C∞(T?H1) given by the Heisenberg structure is h(p,q) =1

2??p,X(q)?2+?p,Y(q)?2?=12(p2x+p2y),(3.1)

where (p,q)?T?H1,?·,·?denotes the duality between covectors and vectors, andp= p xdx+pydy+pωω. See, e.g., [4]. Let us denote by

2h(·) =?p2x+p2y(3.2)

the induced seminorm onT?H1. Then, the sub-Riemannian Dirichlet energyQis the closed form onL2(H1) with coreC∞c(H1), defined by

Q(u) :=?

H

1|du(q)|2dq,?u?C∞c(H1).

Hereddenotes the exterior differential, anddqis the usual Lebesgue measure onR3. The associated diffusion operator is the Heisenberg sub-Laplacian-Δ :=X?X+Y?Y, which in coordinates (x,y,z)?H1reads

Δ =∂2x+∂2y+x2+y2

9

This is a self-adjoint operator inL2(H1).

The form domain ofQis the horizontal Sobolev spaceW1(H1), whileW2(H1) is the domain of the sub-Laplacian. These spaces coincide with theset ofL2(H1) functions such that the following respective norms, computed in the sense of distributions, are finite (see, e.g., [21, 30]): ?u?W1(H1):=?

3.3. Cylindrical coordinates

LetZ:={x=y= 0} ?H1be the center of the group as above, and consider cylindrical coordinates (r,?,z)?R+×S1×RwithR+:= (0,∞), so that for any (x,y,z)? H

1\Z, we write (x,y,z) = (rcos?,rsin?,z). In these coordinates, up to an orthogonal

transformation in the (x,y) coordinates, the basis{dx,dy,ω}ofT?H1is transformed in {dr,rd?,ω}, where

ω=dz-r2

2d?.(3.4)

The corresponding dual basis forTH1is then

R:=∂r,Φ :=1

r∂?+r2∂z,andZ:=∂z.(3.5) That is,{R,Φ}is a global orthonormal frame forD. In these coordinates, the Laplacian (with an abuse of notation denoted by the same symbol Δ) acts on the Hilbert space L

2(R+×S1×R, rdrd?dz) as

Δ =-R?R-Φ?Φ =∂2r+1

r∂r+1r2? ?+r22∂z? 2 .(3.6)

4. Magnetic fields in the Heisenberg group

To motivate the definition of magnetic fields in the case of theHeisenberg group, we start by recalling some facts about magnetic fields in Riemannian geometry.

4.1. Riemannian magnetic fields

A magnetic fieldBon a Riemannian manifold (M,g) is a closed smooth 2-form. Locally, it is always possible to find a 1-formA?Ω1(M) such thatdA=B, known as vector potential forB, thanks to the identification of 1-forms and vector fields induced by the Riemannian metric. Recalling that the Riemannian Hamiltonianh?C∞(T?M) is obtained by duality with the metricg, the action of a magnetic field can then be interpreted as a change in the Hamiltonian functionh?→hA, which reads h

A(p,q) =h(p+A(q),q).(4.1)

10 The corresponding magnetic Laplacian ΔAis obtained by an appropriate quantization ofhA, which yields the associated form (recall (3.2)) Q

A(u) :=?

M |(d+iA)u|2dq,?u?C∞c(M).(4.2)

Note that, by definition,|du|=|?u|:=?

g(?u,?u), where?is the Riemannian gradient. For this reason, the above expression is sometimes written by replacingd with?, implicitly identifying the 1-formAwith the associated vector field.

4.1.1. Gauge invariance

It is clear that if a globally defined vector potentialAforBexists, it is not unique, as A+?is also a vector potential forBas soon as??Ω1(M) is closed. It can be shown that classically this does not pose any problem, as the resulting magnetic trajectories (i.e., projections onMof trajectories onT?Mof integral curves of the Hamiltonian vector field induced byhA) depend only onB.quotesdbs_dbs27.pdfusesText_33

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