DOCSLIB.ORG

The Qft Notes 5

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Qft Notes 5 THE QFT NOTES 5 Badis Ydri Department of Physics, Faculty of Sciences, Annaba University, Annaba, Algeria. December 9, 2011 Contents 1 The Electromagnetic Field 2 1.1 Covariant Formulation of Classical Electrodynamics . ...... 2 1.2 GaugePotentialsandGaugeTransformations . ..... 5 1.3 Maxwell’s Lagrangian Density . 6 1.4 PolarizationVectors ................................. 8 1.5 Quantization of The Electromagnetic Gauge Field . .... 10 1.6 Gupta-BleulerMethod ................................ 15 1.7 Propagator ...................................... 18 1.8 ProblemsandExercises............................... 20 1 1 The Electromagnetic Field 1.1 Covariant Formulation of Classical Electrodynamics The Field Tensor The electric and magnetic fields E~ and B~ generated by a charge density ρ and a current density J~ are given by the Maxwell’s equations written in the Heaviside-Lorentz system as ∇~ E~ = ρ , Gauss′ s Law. (1) ∇~ B~ =0 , No − Magnetic Monopole Law. (2) 1 ∂B~ ∇~ × E~ = − , Faraday′ s Law. (3) c ∂t 1 ∂E~ ∇~ × B~ = (J~ + ) , Ampere − Maxwell′ s Law. (4) c ∂t The Lorentz force law expresses the force exerted on a charge q moving with a velocity ~u in the presence of an electric and magnetic fields E~ and B~ . This is given by 1 F~ = q(E~ + ~u × B~ ). (5) c The continuity equation expresses local conservation of the electric charge. It reads ∂ρ + ∇~ J~ =0. (6) ∂t We consider now the following Lorentz transformation ′ x = γ(x − vt) ′ y = y ′ z = z ′ v t = γ(t − x). (7) c2 In other words (with x0 = ct, x1 = x, x2 = y, x3 = z and signature (+ − −−)) γ −γβ 0 0 µ′ µ ν −γβ γ 0 0 x =Λ νx , Λ= . (8) 0 0 10 0 0 01 2 The transformation laws of the electric and magnetic fields E~ and B~ under this Lorentz trans- formation are given by ′ ′ v ′ v E = E , E = γ(E − B ) , E = γ(E + B ) x x y y c z z z c y ′ ′ v ′ v B = B , B = γ(B + E ) , B = γ(B − E ). (9) x x y y c z z z c y Clearly E~ and B~ do not transform like the spatial part of a 4−vector. In fact E~ and B~ are the components of a second-rank antisymmetric tensor. Let us recall that a second-rank tensor F µν is an abject carrying two indices which transforms under a Lorentz transformation Λ as µν′ µ ν λσ F =Λ λΛ σF . (10) This has 16 components. An antisymmetric tensor will satisfy the extra condition Fµν = −Fµν so the number of independent components is reduced to 6. Explicitly we write 0 F 01 F 02 F 03 01 12 13 µν −F 0 F F F = 02 12 23 . (11) −F −F 0 F −F 03 −F 13 −F 23 0 The transformation laws (10) can then be rewritten as ′ ′ ′ F 01 = F 01 , F 02 = γ(F 02 − βF 12) , F 03 = γ(F 03 + βF 31) ′ ′ ′ F 23 = F 23 , F 31 = γ(F 31 + βF 03) , F 12 = γ(F 12 − βF 02). (12) By comparing (9) and (12) we obtain 01 02 03 12 31 23 F = −Ex , F = −Ey , F = −Ez , F = −Bz , F = −By , F = −Bx. (13) Thus 0 −Ex −Ey −Ez E 0 −B B F µν = x z y . (14) Ey Bz 0 −Bx E −B B 0 z y x Let us remark that (9) remains unchanged under the duality transformation E~ −→ B,~ B~ −→ −E.~ (15) The tensor (14) changes under the above duality transformation to the tensor 0 −Bx −By −Bz B 0 E −E F˜µν = x z y . (16) By −Ez 0 Ex B E −E 0 z y x 3 It is not difficult to show that 1 F˜µν = ǫµναβ F αβ. (17) 2 The 4−dimensional Levi-Civita antisymmetric tensor ǫµναβ is defined in an obvious way. The second-rank antisymmetric tensor F˜ is called the field tensor while the second-rank antisymmetric tensor F˜ is called the dual field tensor. Covariant Formulation The proper charge density ρ0 is the charge density measured in ′ the inertial reference frame O where the charge is at rest. This is given by ρ0 = Q/V0 where V0 is the proper volume. Because the dimension along the direction of the motion is Lorentz 2 2 contracted the volume V measured in the reference frame O is given by V = 1 − u /c V0. Thus the charge density measured in O is q Q ρ ρ = = 0 . (18) V u2 1 − c2 q The current density J~ measured in O is proportional to the velocity ~u and to the current density ρ, viz ρ ~u J~ = ρ~u = 0 . (19) u2 1 − c2 q The 4−vector velocity ηµ is defined by 1 ηµ = (c, ~u). (20) u2 1 − c2 q Hence we can define the current density 4−vector J µ by µ µ J = ρ0η =(cρ,Jx,Jy,Jz). (21) The continuity equation ∇~ J~ = −∂ρ/∂t which expresses charge conservation will take the form µ ∂µJ =0. (22) In terms of Fµν and F˜µν Maxwell’s equations will take the form 1 ∂ F µν = J ν , ∂ F˜µν =0. (23) µ c µ The first equation yields Gauss’s and Ampere-Maxwell’s laws whereas the second equation yields Maxwell’s third equation ∇~ B~ = 0 and Faraday’s law. It remains to write down a covariant Lorentz force. We start with the 4−vector proper force given by q Kµ = η F µν . (24) c ν 4 This is called the Minkowski force. The spatial part of this force is q 1 K~ = (E~ + ~u × B~ ). (25) u2 c 1 − c2 q We have also dpµ Kµ = . (26) dτ In other words d~p dt 1 K~ = = F~ = F.~ (27) dτ dτ u2 1 − c2 q This leads precisely to the Lorentz force law 1 F~ = q(E~ + ~u × B~ ). (28) c 1.2 Gauge Potentials and Gauge Transformations The electric and magnetic fields E~ and B~ can be expressed in terms of a scalar potential V and a vector potential A~ as B~ = ∇~ × A.~ (29) 1 ∂A~ E~ = − (∇~ V + ). (30) c ∂t We construct the 4−vector potential Aµ as Aµ =(V/c, A~). (31) The field tensor Fµν can be rewritten in terms of Aµ as Fµν = ∂µAν − ∂ν Aµ. (32) This equation is actually equivalent to the two equations (29) and (30). The homogeneous µν Maxwell’s equation ∂µF˜ = 0 is automatically solved by this ansatz. The inhomogeneous µν ν Maxwell’s equation ∂µF = J /c becomes 1 ∂ ∂µAν − ∂ν ∂ Aµ = J ν. (33) µ µ c We have a gauge freedom in choosing Aµ given by local gauge transformations of the form (with λ any scalar function) ′ Aµ −→ A µ = Aµ + ∂µλ. (34) 5 Indeed under this transformation we have ′ F µν −→ F µν = F µν . (35) These local gauge transformations form a (gauge) group. In this case the group is just the abelian U(1) unitary group. The invariance of the theory under these transformations is termed a gauge invariance. The 4−vector potential Aµ is called a gauge potential or a gauge field. We make use of the invariance under gauge transformations by working with a gauge potential Aµ which satisfies some extra conditions. This procedure is known as gauge fixing. Some of the gauge conditions so often used are µ ∂µA =0 , Lorentz Gauge. (36) i ∂iA =0 , Coulomb Gauge. (37) A0 =0 , Temporal Gauge. (38) A3 =0 , Axial Gauge. (39) In the Lorentz gauge the equations of motion (33) become 1 ∂ ∂µAν = J ν . (40) µ c µ ′µ µ µ µ Clearly we still have a gauge freedom A −→ A = A +∂ φ where ∂µ∂ φ = 0. In other words if µ µ ′µ ′µ A satisfies the Lorentz gauge ∂µA = 0 then A will also satisfy the Lorentz gauge, i.e. ∂µA = µ 0 iff ∂µ∂ φ = 0. This residual gauge symmetry can be fixed by imposing another condition such as the temporal gauge A0 = 0. We have therefore 2 constraints imposed on the components of the gauge potential Aµ which means that only two of them are really independent. 1.3 Maxwell’s Lagrangian Density The equations of motion of the gauge field Aµ is 1 ∂ ∂µAν − ∂ν ∂ Aµ = J ν. (41) µ µ c These equations of motion should be derived from a local Lagrangian density L, i.e. a La- grangian which depends only on the fields and their first derivatives at the point ~x. We have then L = L(Aµ, ∂νAµ). (42) The Lagrangian is the integral over ~x of the Lagrangian density, viz L = d~xL. (43) Z 6 The action is the integral over time of L, namely S = dtL = d4xL. (44) Z Z We compute δS = d4xδL Z 4 δL δL δL = d x δAν − δAν∂µ + ∂µ δAν . (45) Z δAν δ∂µAν δ∂µAν The surface term is zero because the field Aν at infinity is assumed to be zero and thus µ δAν =0 , x −→ ±∞. (46) We get 4 δL δL δS = d xδAν − ∂µ . (47) Z δAν δ∂µAν The principle of least action δS = 0 yields therefore the Euler-Lagrange equations δL δL − ∂µ =0. (48) δAν δ∂µAν Firstly the Lagrangian density L is a Lorentz scalar. Secondly the equations of motion (41) are linear in the field Aµ and hence the Lagrangian density L can at most be quadratic in Aµ. The most general form of L which is quadratic in Aµ is µ 2 ν µ ν µ µ µ LMaxwell = α(∂µA ) + β(∂µA )(∂ Aν )+ γ(∂µA )(∂ν A )+ δAµA + ǫJµA .
Load more

Recommended publications
  • Quantum Gravity and the Cosmological Constant Problem
    Quantum Gravity and the Cosmological Constant Problem J. W. Moffat Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada and Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada October 15, 2018 Abstract A finite and unitary nonlocal formulation of quantum gravity is applied to the cosmological constant problem. The entire functions in momentum space at the graviton-standard model particle loop vertices generate an exponential suppression of the vacuum density and the cosmological constant to produce agreement with their observational bounds. 1 Introduction A nonlocal quantum field theory and quantum gravity theory has been formulated that leads to a finite, unitary and locally gauge invariant theory [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. For quantum gravity the finiteness of quantum loops avoids the problem of the non-renormalizabilty of local quantum gravity [15, 16]. The finiteness of the nonlocal quantum field theory draws from the fact that factors of exp[ (p2)/Λ2] are attached to propagators which suppress any ultraviolet divergences in Euclidean momentum space,K where Λ is an energy scale factor. An important feature of the field theory is that only the quantum loop graphs have nonlocal properties; the classical tree graph theory retains full causal and local behavior. Consider first the 4-dimensional spacetime to be approximately flat Minkowski spacetime. Let us denote by f a generic local field and write the standard local Lagrangian as [f]= [f]+ [f], (1) L LF LI where and denote the free part and the interaction part of the action, respectively, and LF LI arXiv:1407.2086v1 [gr-qc] 7 Jul 2014 1 [f]= f f .
    [Show full text]
  • Gauge-Fixing Degeneracies and Confinement in Non-Abelian Gauge Theories
    PH YSICAL RKVIE% 0 VOLUME 17, NUMBER 4 15 FEBRUARY 1978 Gauge-fixing degeneracies and confinement in non-Abelian gauge theories Carl M. Bender Department of I'hysics, 8'ashington University, St. Louis, Missouri 63130 Tohru Eguchi Stanford Linear Accelerator Center, Stanford, California 94305 Heinz Pagels The Rockefeller University, New York, N. Y. 10021 (Received 28 September 1977) Following several suggestions of Gribov we have examined the problem of gauge-fixing degeneracies in non- Abelian gauge theories. First we modify the usual Faddeev-Popov prescription to take gauge-fixing degeneracies into account. We obtain a formal expression for the generating functional which is invariant under finite gauge transformations and which counts gauge-equivalent orbits only once. Next we examine the instantaneous Coulomb interaction in the canonical formalism with the Coulomb-gauge condition. We find that the spectrum of the Coulomb Green's function in an external monopole-hke field configuration has an accumulation of negative-energy bound states at E = 0. Using semiclassical methods we show that this accumulation phenomenon, which is closely linked with gauge-fixing degeneracies, modifies the usual Coulomb propagator from (k) ' to Iki ' for small )k[. This confinement behavior depends only on the long-range behavior of the field configuration. We thereby demonstrate the conjectured confinement property of non-Abelian gauge theories in the Coulomb gauge. I. INTRODUCTION lution to the theory. The failure of the Borel sum to define the theory and gauge-fixing degeneracy It has recently been observed by Gribov' that in are correlated phenomena. non-Abelian gauge theories, in contrast with Abe- In this article we examine the gauge-fixing de- lian theories, standard gauge-fixing conditions of generacy further.
    [Show full text]
  • Gauge Symmetry Breaking in Gauge Theories---In Search of Clarification
    Gauge Symmetry Breaking in Gauge Theories—In Search of Clarification Simon Friederich [email protected] Universit¨at Wuppertal, Fachbereich C – Mathematik und Naturwissenschaften, Gaußstr. 20, D-42119 Wuppertal, Germany Abstract: The paper investigates the spontaneous breaking of gauge sym- metries in gauge theories from a philosophical angle, taking into account the fact that the notion of a spontaneously broken local gauge symmetry, though widely employed in textbook expositions of the Higgs mechanism, is not supported by our leading theoretical frameworks of gauge quantum the- ories. In the context of lattice gauge theory, the statement that local gauge symmetry cannot be spontaneously broken can even be made rigorous in the form of Elitzur’s theorem. Nevertheless, gauge symmetry breaking does occur in gauge quantum field theories in the form of the breaking of remnant subgroups of the original local gauge group under which the theories typi- cally remain invariant after gauge fixing. The paper discusses the relation between these instances of symmetry breaking and phase transitions and draws some more general conclusions for the philosophical interpretation of gauge symmetries and their breaking. 1 Introduction The interpretation of symmetries and symmetry breaking has been recog- nized as a central topic in the philosophy of science in recent years. Gauge symmetries, in particular, have attracted a considerable amount of inter- arXiv:1107.4664v2 [physics.hist-ph] 5 Aug 2012 est due to the central role they play in our most successful theories of the fundamental constituents of nature. The standard model of elementary par- ticle physics, for instance, is formulated in terms of gauge symmetry, and so are its most discussed extensions.
    [Show full text]
  • Operator Gauge Symmetry in QED
    Symmetry, Integrability and Geometry: Methods and Applications Vol. 2 (2006), Paper 013, 7 pages Operator Gauge Symmetry in QED Siamak KHADEMI † and Sadollah NASIRI †‡ † Department of Physics, Zanjan University, P.O. Box 313, Zanjan, Iran E-mail: [email protected] URL: http://www.znu.ac.ir/Members/khademi.htm ‡ Institute for Advanced Studies in Basic Sciences, IASBS, Zanjan, Iran E-mail: [email protected] Received October 09, 2005, in final form January 17, 2006; Published online January 30, 2006 Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper013/ Abstract. In this paper, operator gauge transformation, first introduced by Kobe, is applied to Maxwell’s equations and continuity equation in QED. The gauge invariance is satisfied after quantization of electromagnetic fields. Inherent nonlinearity in Maxwell’s equations is obtained as a direct result due to the nonlinearity of the operator gauge trans- formations. The operator gauge invariant Maxwell’s equations and corresponding charge conservation are obtained by defining the generalized derivatives of the f irst and second kinds. Conservation laws for the real and virtual charges are obtained too. The additional terms in the field strength tensor are interpreted as electric and magnetic polarization of the vacuum. Key words: gauge transformation; Maxwell’s equations; electromagnetic fields 2000 Mathematics Subject Classification: 81V80; 78A25 1 Introduction Conserved quantities of a system are direct consequences of symmetries inherent in the system. Therefore, symmetries are fundamental properties of any dynamical system [1, 2, 3]. For a sys- tem of electric charges an important quantity that is always expected to be conserved is the total charge.
    [Show full text]
  • Avoiding Gauge Ambiguities in Cavity Quantum Electrodynamics Dominic M
    www.nature.com/scientificreports OPEN Avoiding gauge ambiguities in cavity quantum electrodynamics Dominic M. Rouse1*, Brendon W. Lovett1, Erik M. Gauger2 & Niclas Westerberg2,3* Systems of interacting charges and felds are ubiquitous in physics. Recently, it has been shown that Hamiltonians derived using diferent gauges can yield diferent physical results when matter degrees of freedom are truncated to a few low-lying energy eigenstates. This efect is particularly prominent in the ultra-strong coupling regime. Such ambiguities arise because transformations reshufe the partition between light and matter degrees of freedom and so level truncation is a gauge dependent approximation. To avoid this gauge ambiguity, we redefne the electromagnetic felds in terms of potentials for which the resulting canonical momenta and Hamiltonian are explicitly unchanged by the gauge choice of this theory. Instead the light/matter partition is assigned by the intuitive choice of separating an electric feld between displacement and polarisation contributions. This approach is an attractive choice in typical cavity quantum electrodynamics situations. Te gauge invariance of quantum electrodynamics (QED) is fundamental to the theory and can be used to greatly simplify calculations1–8. Of course, gauge invariance implies that physical observables are the same in all gauges despite superfcial diferences in the mathematics. However, it has recently been shown that the invariance is lost in the strong light/matter coupling regime if the matter degrees of freedom are treated as quantum systems with a fxed number of energy levels8–14, including the commonly used two-level truncation (2LT). At the origin of this is the role of gauge transformations (GTs) in deciding the partition between the light and matter degrees of freedom, even if the primary role of gauge freedom is to enforce Gauss’s law.
    [Show full text]
  • 8 Non-Abelian Gauge Theory: Perturbative Quantization
    8 Non-Abelian Gauge Theory: Perturbative Quantization We’re now ready to consider the quantum theory of Yang–Mills. In the first few sec- tions, we’ll treat the path integral formally as an integral over infinite dimensional spaces, without worrying about imposing a regularization. We’ll turn to questions about using renormalization to make sense of these formal integrals in section 8.3. To specify Yang–Mills theory, we had to pick a principal G bundle P M together ! with a connection on P . So our first thought might be to try to define the Yang–Mills r partition function as ? SYM[ ]/~ Z [(M,g), g ] = A e− r (8.1) YM YM D ZA where 1 SYM[ ]= tr(F F ) (8.2) r −2g2 r ^⇤ r YM ZM as before, and is the space of all connections on P . A To understand what this integral might mean, first note that, given any two connections and , the 1-parameter family r r0 ⌧ = ⌧ +(1 ⌧) 0 (8.3) r r − r is also a connection for all ⌧ [0, 1]. For example, you can check that the rhs has the 2 behaviour expected of a connection under any gauge transformation. Thus we can find a 1 path in between any two connections. Since 0 ⌦M (g), we conclude that is an A r r2 1 A infinite dimensional affine space whose tangent space at any point is ⌦M (g), the infinite dimensional space of all g–valued covectors on M. In fact, it’s easy to write down a flat (L2-)metric on using the metric on M: A 2 1 µ⌫ a a d ds = tr(δA δA)= g δAµ δA⌫ pg d x.
    [Show full text]
  • Spontaneous Symmetry Breaking in the Higgs Mechanism
    Spontaneous symmetry breaking in the Higgs mechanism August 2012 Abstract The Higgs mechanism is very powerful: it furnishes a description of the elec- troweak theory in the Standard Model which has a convincing experimental ver- ification. But although the Higgs mechanism had been applied successfully, the conceptual background is not clear. The Higgs mechanism is often presented as spontaneous breaking of a local gauge symmetry. But a local gauge symmetry is rooted in redundancy of description: gauge transformations connect states that cannot be physically distinguished. A gauge symmetry is therefore not a sym- metry of nature, but of our description of nature. The spontaneous breaking of such a symmetry cannot be expected to have physical e↵ects since asymmetries are not reflected in the physics. If spontaneous gauge symmetry breaking cannot have physical e↵ects, this causes conceptual problems for the Higgs mechanism, if taken to be described as spontaneous gauge symmetry breaking. In a gauge invariant theory, gauge fixing is necessary to retrieve the physics from the theory. This means that also in a theory with spontaneous gauge sym- metry breaking, a gauge should be fixed. But gauge fixing itself breaks the gauge symmetry, and thereby obscures the spontaneous breaking of the symmetry. It suggests that spontaneous gauge symmetry breaking is not part of the physics, but an unphysical artifact of the redundancy in description. However, the Higgs mechanism can be formulated in a gauge independent way, without spontaneous symmetry breaking. The same outcome as in the account with spontaneous symmetry breaking is obtained. It is concluded that even though spontaneous gauge symmetry breaking cannot have physical consequences, the Higgs mechanism is not in conceptual danger.
    [Show full text]
  • Gauge-Independent Higgs Mechanism and the Implications for Quark Confinement
    EPJ Web of Conferences 137, 03009 (2017) DOI: 10.1051/ epjconf/201713703009 XIIth Quark Confinement & the Hadron Spectrum Gauge-independent Higgs mechanism and the implications for quark confinement Kei-Ichi Kondo1;a 1Department of Physics, Faculty of Science, Chiba University, Chiba 263-8522, Japan Abstract. We propose a gauge-invariant description for the Higgs mechanism by which a gauge boson acquires the mass. We do not need to assume spontaneous breakdown of gauge symmetry signaled by a non-vanishing vacuum expectation value of the scalar field. In fact, we give a manifestly gauge-invariant description of the Higgs mechanism in the operator level, which does not rely on spontaneous symmetry breaking. For concrete- ness, we discuss the gauge-Higgs models with U(1) and SU(2) gauge groups explicitly. This enables us to discuss the confinement-Higgs complementarity from a new perspec- tive. 1 Introduction Spontaneous symmetry breaking (SSB) is an important concept in physics. It is known that SSB occurs when the lowest energy state or the vacuum is degenerate [1]. First, we consider the SSB of the global continuous symmetry G = U(1). The complex scalar field theory described by the following Lagrangian density Lcs has the global U(1) symmetry: ! 2 2 ∗ µ ∗ ∗ λ ∗ µ L = @µϕ @ ϕ − V(ϕ ϕ); V(ϕ ϕ) = ϕ ϕ − ; ϕ 2 C; λ > 0; (1) cs 2 λ where ∗ denotes the complex conjugate and the classical stability of the theory requires λ > 0. Indeed, this theory has the global U(1) symmetry, since Lcs is invariant under the global U(1) transformation, i.e., the phase transformation: ϕ(x) ! eiθϕ(x): (2) If the potential has the unique minimum (µ2 ≤ 0) as shown in the left panel of Fig.
    [Show full text]
  • Gauge Fixing and Renormalization in Topological Quantum Field Theory*
    SLAC-PUB-4630 May 1988 P3 Gauge Fixing and Renormalization in Topological Quantum Field Theory* ROGER BROOKS, DAVID MONTANO, AND JACOB SONNENSCHEIN+ Stanford Linear Accelerator Center Stanford University, Stanford, California 94 309 ABSTRACT In this letter we show how Witten’s topological Yang-Mills and gravitational quantum field theories may be obtained by a straightforward BRST gauge fixing procedure. We investigate some aspects of the renormalization of the topological Yang-Mills theory. It is found that the beta function for the Yang-Mills coupling constant is not zero. Submitted to Phys. Lett. * Work supported by the Department of Energy, contract DE-AC03-76SF00515. + Work supported by Dr. Chaim Weismann Postdoctoral Fellowship. 1. INTRODUCTION Recently, there has been an effort to study the topology of four dimensional manifolds using the methods of quantum field theory!‘-‘] This was motivated by Donaldson’s use of self-dual Yang-Mills equations to investigate the topology of low dimensional manifolds!” The work of Floer on three manifolds, as interpreted by Atiyah!“can be understood as a modified non-relativistic field theory. Atiyah then conjectured that a relativistic quantum field theory could be used to study Donaldson’s invariants on four manifolds. This led Witten to work out a series of preprints in which just such topological quantum field theories (TQFT) were described.14-61 In the first paper a TQFT was proposed whose correlation func- tions were purely topological and reproduced Donaldson’s invariants!Q’ A later paper extended these ideas to a more complicated Lagrangian which could be used to study similar invariants in topological gravity!51These TQFT’s all pos- sess a fermionic symmetry analogous to a BRST symmetry.
    [Show full text]
  • Gauge-Underdetermination and Shades of Locality in the Aharonov-Bohm Effect*
    Gauge-underdetermination and shades of locality in the Aharonov-Bohm effect* Ruward A. Mulder Abstract I address the view that the classical electromagnetic potentials are shown by the Aharonov-Bohm effect to be physically real (which I dub: `the potentials view'). I give a historico-philosophical presentation of this view and assess its prospects, more precisely than has so far been done in the literature. Taking the potential as physically real runs prima facie into `gauge-underdetermination': different gauge choices repre- sent different physical states of affairs and hence different theories. This fact is usually not acknowledged in the literature (or in classrooms), neither by proponents nor by opponents of the potentials view. I then illustrate this theme by what I take to be the basic insight of the AB effect for the potentials view, namely that the gauge equivalence class that directly corresponds to the electric and magnetic fields (which I call the Wide Equivalence Class) is too wide, i.e., the Narrow Equivalence Class encodes additional physical degrees of freedom: these only play a distinct role in a multiply-connected space. There is a trade-off between explanatory power and gauge symmetries. On the one hand, this narrower equivalence class gives a local explanation of the AB effect in the sense that the phase is incrementally picked up along the path of the electron. On the other hand, locality is not satisfied in the sense of signal locality, viz. the finite speed of propagation exhibited by electric and magnetic fields. It is therefore intellec- tually mandatory to seek desiderata that will distinguish even within these narrower equivalence classes, i.e.
    [Show full text]
  • Gauge Theory
    Preprint typeset in JHEP style - HYPER VERSION 2018 Gauge Theory David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/gaugetheory.html [email protected] Contents 0. Introduction 1 1. Topics in Electromagnetism 3 1.1 Magnetic Monopoles 3 1.1.1 Dirac Quantisation 4 1.1.2 A Patchwork of Gauge Fields 6 1.1.3 Monopoles and Angular Momentum 8 1.2 The Theta Term 10 1.2.1 The Topological Insulator 11 1.2.2 A Mirage Monopole 14 1.2.3 The Witten Effect 16 1.2.4 Why θ is Periodic 18 1.2.5 Parity, Time-Reversal and θ = π 21 1.3 Further Reading 22 2. Yang-Mills Theory 26 2.1 Introducing Yang-Mills 26 2.1.1 The Action 29 2.1.2 Gauge Symmetry 31 2.1.3 Wilson Lines and Wilson Loops 33 2.2 The Theta Term 38 2.2.1 Canonical Quantisation of Yang-Mills 40 2.2.2 The Wavefunction and the Chern-Simons Functional 42 2.2.3 Analogies From Quantum Mechanics 47 2.3 Instantons 51 2.3.1 The Self-Dual Yang-Mills Equations 52 2.3.2 Tunnelling: Another Quantum Mechanics Analogy 56 2.3.3 Instanton Contributions to the Path Integral 58 2.4 The Flow to Strong Coupling 61 2.4.1 Anti-Screening and Paramagnetism 65 2.4.2 Computing the Beta Function 67 2.5 Electric Probes 74 2.5.1 Coulomb vs Confining 74 2.5.2 An Analogy: Flux Lines in a Superconductor 78 { 1 { 2.5.3 Wilson Loops Revisited 85 2.6 Magnetic Probes 88 2.6.1 't Hooft Lines 89 2.6.2 SU(N) vs SU(N)=ZN 92 2.6.3 What is the Gauge Group of the Standard Model? 97 2.7 Dynamical Matter 99 2.7.1 The Beta Function Revisited 100 2.7.2 The Infra-Red Phases of QCD-like Theories 102 2.7.3 The Higgs vs Confining Phase 105 2.8 't Hooft-Polyakov Monopoles 109 2.8.1 Monopole Solutions 112 2.8.2 The Witten Effect Again 114 2.9 Further Reading 115 3.
    [Show full text]
  • Path Integral and Gauge Fixing
    11.12.2020 Non-abelian gauge theories: path integral and gauge fixing (Lecture notes - a.a. 2020/21) Fiorenzo Bastianelli 1 Faddeev-Popov The quantization of gauge theories requires additional considerations for constructing a well- defined QFT and its perturbative expansion. The problem is clear in the path integral approach, where a naive path integral quantization gives a diverging result. Let us reconsider the case of the abelian U(1) theory (free electromagnetism) Z 1 S[A] = d4x − F F µν 4 µν Z (1) Z = DA eiS[A] ∼ 1 where Fµν = @µAν − @νAµ. Here the path integral diverges because one is summing over an infinite number of gauge equivalent configurations g −1 iα(x) Aµ(x) ! Aµ(x) = Aµ(x) + ig(x)@µg (x) ; g(x) = e 2 U(1) (2) which have the same value of the action, S[Ag] = S[A]. The field space decomposes into inequivalent gauge orbits, as in figure Figure 1: Gauge orbits One would like to define the path integral in such a way of getting a finite and gauge invariant result Z DA Z = eiS[A] ∼ finite : (3) Vol(Gauge) This can be done using a gauge fixing function `ala Faddeev-Popov, where unphysical ghost fields are introduced to exponentiate a measure factor. Ideally, the gauge fixing function should pick just one representative from each gauge orbits, as sketched in figure 2 1 Figure 2: Gauge fixing by the condition f(A) = 0 The gauge-fixed action thus obtained exhibits a global symmetry, called BRST symmetry, which is a remnant of the gauge symmetry and which is very powerful to prove many prop- erties of quantum gauge theories.
    [Show full text]