Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Point particle contact interactions and Coulomb’s law James P. Edwards IFM Morelia Nov 2016 Based on [arXiv:1506.08130 [hep-th]] and [arXiv:1409.4948 [hep-th]] Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Outline Hola! 1 Introduction 2 Worldline formalism Electric field and Coulomb’s law 3 Faraday’s lines of flux Finite T Results Generalisations 4 Spin 1/2 particles Relativistic contact interactions Conclusion 5 Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Introducci´ on... Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Introducci´ on... ...de mi!! Acabo de empezar una beca posdoctoral aqu´ ı en el IFM – estoy muy contento de haber ingresado en el grupo! Ojal´ a y lleguemos a conocernos muy bien en el futuro... Mi trabajo se trata del formalismo linea de mundo (worldline formalism) de los campos cu´ anticos. Particularmente he trabajado en: Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Introducci´ on... ...de mi!! Acabo de empezar una beca posdoctoral aqu´ ı en el IFM – estoy muy contento de haber ingresado en el grupo! Ojal´ a y lleguemos a conocernos muy bien en el futuro... Mi trabajo se trata del formalismo linea de mundo (worldline formalism) de los campos cu´ anticos. Particularmente he trabajado en: La din´ amica de espin cu´ anitico de los electrones para las computadoras cu´ anticas University of Oxford 2009 La ecuaci´ on Wheeler-deWitt (Restricci´ on de Hamiltonian en relatividad general) University of Cambridge 2009-2011 Interacciones de contacto dentro de la teor´ ıa de spinning strings sin tensi´ on University of Durham 2011-2015 La anomal´ ıa de la simmetr´ ıa de Weyl y la teor´ ıa de campo de Liouville en dimensiones D < 26 El modelo est´ andar de part´ ıculas y teor´ ıas unificadas SU (5) , SO (10) y SO (16) . El formalismo linea de mundo University of Durham 2012-2015 y University of Bath 2015-2016 La ley de Coulomb y la incorporaci´ on de espin Las teor´ ıas non-Abelian en este formalismo (la interacci´ on de Wilson) Los campos cu´ anticos en el espacio no-conmutativo. Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Trabajo actual Mi trabajo en Morelia es en colaboraci´ on con Christian Schubert y otras personas que dependen del proyecto: Los propagadores en vac´ ıo y vestidos en campos constantes en el formalismo linea del mundo Fiorenzo Bastianelli (Bologna), Olindo Corradini (Modena), Naser Ahmadiniaz (Institute for Basic Science) La tear´ ıa U ( N ) en el espacio no-conmutativo con campos de color en la linea de mundo. Olindo Corradini, Pablo Pisani (La Plata), Naser Ahmadiniaz, Idrish Huet (UNACH) C´ alculos num´ ericos de las lineas de mundo – estados ligardos en campos cu´ anticos y mec´ anica cu´ antica Axel Weber (IFM), Anabel Trejo (IFM), Urs Gerber (UNAM) Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Electric field and Coulomb’s law Worldline formalism Generalisations Conclusion Worldline formalism The so-called worldline formalism of quantum field theory is a first quantised approach originally motivated by the high tension limit of string theory. It offers significant computational advantages over the conventional perturbative methods traditionally employed in field theory. It enjoys calculational efficiency and compactness. Feynman diagrams are combined together into gauge invariant combinations. There are indications of a close relationship between string theory, particle worldlines and field theory. Has a very nice physical interpretation. k 1 + p p k µ k ν p 1 2 k µ k ν 1 1 1 k µ k µ 1 (a) Two diagrams represent a process in scalar quantum (b) Worldline techniques require field theory. only a single diagram. Figure: Feynman diagrams in field theory and the worldline formalism Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Electric field and Coulomb’s law Worldline formalism Generalisations Conclusion Worldline formalism In the worldline formalism, the basic idea is to integrate out the matter degrees of freedom. The resulting functional determinant is then recast as a transition amplitude in quantum mechanics [ 1 ] . For example, for spinor quantum electrodynamics ( D ≡ ∂ + igA ) �� � � � � � ¯ � d 4 x ¯ Γ Ψ [ A ] = log D ΨΨ exp − Ψ ( γ · D − m ) Ψ � ( γ · D ) 2 + m 2 � = − 1 2Tr log � ∞ � � dT T e − m 2 T D ψ e − S [ x,ψ ] , − → D x 0 x (0)= x ( T ) ψ (0)= − ψ ( T ) where � ˙ � � T x 2 4 + 1 2 ψ · ˙ x − ieψ µ F µν ( x ) ψ ν S [ x, ψ ] = dτ ψ + ieA ( x ) · ˙ (1) 0 1 Strassler, Nucl. Phys. B385 Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Electric field and Coulomb’s law Worldline formalism Generalisations Conclusion Applications Worldline approaches have been used successfully to address many problems, including N-point scattering amplitudes at one loop order, extended to incorporate multi-loop diagrams. Determination of effective actions Higher spin fields Graviton / photon production Trace anomalies Non-commutative quantum field theory Grand unification Localisation Landau-Khalatnikov-Fradkin transformations and dressed propagators. Gravitational anomalies Non-Abelian field theory. Worldline approaches are steadily growing in popularity and have an enormous potential to aid our understanding of fundamental interactions between particles. Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Electric field and Coulomb’s law Worldline formalism Generalisations Conclusion Higher order interactions Feynman originally gave the general action for the exchange of an arbitrary number of gauge bosons – for tree level processes in a scalar QED: � ∞ � x ( T )= x 1 � � � T x 2 − ie 2 � T � T 0 dτ ′ ˙ ˙ x µ ∆ µν ( x − x ′ ) ˙ x ′ ν − 0 dτ 4 + ieA ( x ) · ˙ x 0 dτ dTe − m 2 T 2 D xe (2) 0 x (0)= x 0 This motivates us to study different interactions on the particle worldline. In particular we will consider the infinite mass limit of the propagator and replace � � → gδ µν δ D � � e 2 ∆ µν x ( τ ) − x ( τ ′ ) x ( τ ) − x ( τ ′ ) − (3) where g absorbs the mass dependence through g = e 2 µ 2 Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Faraday’s lines of flux Electric field and Coulomb’s law Finite T Generalisations Results Conclusion Particle worldlines Consider a pair of equal and oppositely charged particles at the spatial points a and b . Note that � dτ d ω dτ δ 3 ( ω ( τ ) − x ) E ′ ( x ) = q (4) C satisfies Gauss’ law: ∇ · E ′ = qδ 3 ( x − a ) − qδ 3 ( x − b ) . It does not, however, satisfy ∇ × E ′ = 0 , so it is not (yet!) a physical electric field. So we examine its functional average over curves, C , � ω ( T )= b � � � � T dτ d ω ω 2 dτ δ 3 ( ω ( τ ) − x ) e − ˙ E ′ ( x ) 4 dτ T = q D ω (5) 0 ω (0)= a C In Fourier space the insertion is familiar to worldline theorists and analogous to the vertex operators of string theory: � T � � � � E ′ i ( k ) w i ( τ ) e i k · ω ( τ ) V i V i T = q k ( τ ) ; k ( τ ) = ˙ (6) dτ 0 Point particle contact interactions and Coulomb’s law James P. Edwards
Hola! Introduction Faraday’s lines of flux Electric field and Coulomb’s law Finite T Generalisations Results Conclusion Interpretation There are complementary ways to interpret this average: One may interpret this action as similar to a thermal average (at temperature T ) � ω 2 ˙ against a Hamiltonian H = 4 dτ . The average is over all trajectories that pass through the spatial point x and whose endpoints are at a and b . Alternatively, one can think of T as setting the intrinsic length measured along the particle worldline This worldline is allowed to fluctuate (with fixed endpoints) and we count only the contribution of those fluctuations that pass through the point x . In the high temperature limit, T → ∞ , the contributions to the average are localised to the endpoints and we find � 1 � � e i k · a − e i k · b � � � T = qi k E ′ ( k ) lim + O (7) k 2 k 2 T T →∞ which in three dimensions gives the classical dipole field at the point x : � � � � − 1 1 = q E ′ ( x ) lim 4 π ∇ | x − a | + (8) . | x − b | T →∞ Point particle contact interactions and Coulomb’s law James P. Edwards
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