N-C from the Noether Procedure and Galilean Electrodynamics

Transcription

1 gen. Vertical version with logotype under the N-C from the Noether Procedure and Galilean Electrodynamics Dennis Hansen 10th Nordic String Theory Meeting, Bremen 2016 The Niels Bohr Institute and Niels Bohr International Academy (Based on [1603.X] and [1603.Y] with G. Festuccia, J. Hartong, N. Obers)

2 Outline Introduction Non-Relativistic Fields and NC Galilean Electrodynamics Summary

3 Introduction

4 Brief Review Figuresof (Torsional) Newton-Cartan 19. februar :53 Newton-Cartan geometry with τ µ, h µν = δ ij e µ i ej ν (see for example [ ], [ ], [ ]). Covariant derivatives contains gauge fields Ωρ i, ωρ i j, Γλ ρµ: ρ τ µ = ρ τ µ Γ λ ρµτ λ = 0 ρ e i µ = ρ e i µ Γ λ ρµe i λ Ω i ρ τ µ ω i ρ je j µ = 0. In TNC there is no Levi-Civita connection [ ].

5 Special TNC connection What is the connection that is most analog to the Levi-Civita connection?

6 Special TNC connection What is the connection that is most analog to the Levi-Civita connection? There exists a unique connection linear in M µ transforming under infinitesimal Galilean boosts λ i and GCTs ξ µ as δm µ = L ξ M µ + e i µλ i. The affine connection of this special TNC connection: Γ λ µν = ˆv λ µ τ ν hλσ ( µ h νσ + ν h µσ σ h µν ) ˆv µ v µ h µλ M λ, h µν h µν 2τ (µ M ν). This is torsionful connection with Tµν = 2ˆv λ [µ τ ν]. It is not clear that this a particularly interesting or useful connection: Noether procedure clears this up. λ

7 Non-Relativistic Fields and NC

8 Conserved Noether Currents of an Action S (0) = d D x L(ϕ, ϕ) A NR (Bargmann) field with mass M transforms as ϕ ( l t,x ) ( = e if (t,x)m S ll ϕ l (t,x), S ll rep SO(d) R d). Noether s theorem and conservation laws for action S (0) : E µ can energy, T can µi, J can µ, momentum mass, only present for M 0 b can µi = tt can µi x i J can µ + w µi boost, j can µij = 2x [i T can µ j] + s µij. ang. mom. The non-conserved spin- and lift-currents s µij, w µi. Conservation laws for boost and rotation currents: µ b µi can = 0 T 0i can = J i can µ w µi µ j µij can = 0 2T [ij] can = µ s µij. Massless NR theories are very different from rel. ones: Symmetry charge densities Tcan 0i = bcan 0i = 0 when w µi = 0.

9 Improvements of Currents Currents are only defined up to a total derivative, i.e. they can be improved as T µi imp = T can µi + ρ B ρµi. Current simplifications and choices leading to maximal simplification, important result: j µij imp = 2x [i T µ j] imp T ij imp = T ji imp M 0 : b µi imp = tt µi imp x i J µ imp Timp 0i = Jimp i M = 0 : b µi imp = tt µi imp + ψµi Timp 0i = µ ψ µi [ ψ µi δ µ 0 w 0i] δ µ j [w [ij] + 1 ] 2 s0ij. ψ µi is non-conserved but is related to improvements Φ µ ρ C ρµ of the mass current J µ : ( Φ µ 0 ψ 0j + i ψ ij) δ µ j + j ψ 0j δ µ 0.

10 Gauging Global Spacetime Symmetries Notice: The variation of the action wrt. local parameters: δs 0 [ϕ] = d D x µ ξ N (x)j (0)µN. Noether procedure: Making global symmetry local by introducing gauge fields: M S [ϕ,a] = S (0) [ϕ] + S (1) [ϕ,a] + S (2) [ϕ,a] +... S (1) [ϕ,a] = M universal! d D x A µn J (0)µN, δ (1) A µn µ ξ N. For a Bargmann or Galilean theory we introduce gauge fields τ µ, e µi, M µ, Ω µi, ω µij with coupling: [ S (1) d D x τ µ E can µ + e µi T can µi M µ J can µ + 1 ] M only Barg. 2 ω µijs µij Ω µi w µi,

11 Gauging Global Bargmann Spacetime Symmetries Express canonical currents in terms of maximally simplifying improvements and do integration by parts: [ S (1) = d D x τ µ E can µ + e 0i Timp 0i + 1 M 2 s ijt ij imp M µj µ imp + 1 ] 2 C µijs µij C µi w µi, v i e 0i, s ij 2e (ij), C µi Ω µi Ω µi, Ω µi δ k µ C µij ω µij ω µij ) 2δµ 0 [0 M i] δµ k [k M i] ( 1 2 0s ki + (k v i) ω µij δ k µ ( [i s j]k k e [ij] ) δ 0 µ ( 0 e [ij] + [i v j] [i M j] ). Extracting e µi,m µ -dependent connection from improvements. Contortions C µi, C µij and minimal coupling.

12 Gauging Global Galilean Spacetime Symmetries No coupling to M µ : Inserting the maximally simplified improved currents is now a different story: [ S (1) = d D x τ µ E can µ +e 0i Timp 0i s ijt ij imp C µijs µij C µi w ], µi M C µi Ω µi ˆΩ µi, C µij ω µij ˆω µij ( ) Ω ˆ 1 µi δµ k 2 0s ki + (k v i) ˆω µij δµ k ( ) ( ) [i s j]k k e [ij] δ 0 µ 0 e [ij] + [i v j]. No good dependent connection from the improvements. How to obtain the minimal connection from field theoretic argument: Constructing an uncoupled current Φ µ from ψ µi. Add coupling M µ Φ µ.

13 Reproducing Newton-Cartan Geometry Identifying gauge fields and vielbeins: Linearization τ µ = δ 0 µ + τ µ + O (2), e i µ = δ i µ + e i µ + O (2). Identifying the pseudo-connection ˆΩ µi, ˆω µij in Galilean theories: No Levi-Civita-like connection. Further Ω µi, ω µij is the special TNC connection and hence we conclude the minimal TNC connection! For the special TNC connection Ω µi, ω µij coupling is always: M µ (J µ can + Φ µ ). However there are other α R connections involving just M µ not predicted by Noether procedure [ ]: Γ λ µν (α) = ˆv λ µ τ ν hλσ ( µ H νσ + ν H µσ σ H µν ) H µν (α) h µν + α Φτ µ τ ν, Φ v µ M µ + ½h µν M µ M ν.

14 Galilean Electrodynamics

15 Galilean Electrodynamics on Flat Spacetime Maxwellian electrodynamics is a relativistic U(1) gauge theory. There are two non-relativistic limits at the level of the Maxwell EOMs [Le Bellac & Lévy-Leblond 1973]: [ ] F = d 3 x ρe electric effects + J B magnetic effects, Null reduction (dimensional reduction along null direction) of MED: Another theory of non-relativistic electrodynamics : L = 1 2 B B + E Ẽ a2 ρ ϕ + J A ρϕ. ED with four field strengths B, E, Ẽ, a expressed in terms of three potentials ϕ, A, ϕ (see [Santos et. al. 2005]): B i ɛ i jk j A k Ẽ i i ϕ t A i E i i ϕ a t ϕ.

16 Galilean Electrodynamics on Curved Spacetimes One can perform a null reduction of MED on a general Lorentzian manifold to get GED on TNC. EOMs: Covariant derivatives are wrt. the special TNC con.: µ Z µ = 2 [µ τ ν]ˆv µ Z ν E µ h µν E ν µ E µ = [µ τ ν] (W µν 2ˆv [µ E ν]) W λµ B λµ + 2E [λ v µ] µ W µρ = [µ τ ν] (2ˆv µ W νρ + ˆv ρ W µν ) Z ρ Ẽ ρ v ρ a M σ W σρ. First to study GED on general TNC (as far as we know): compare to literature [Künzle 1976], [ ]. Sources can be added.

17 Non-Relativistic Scalar QED Perform a null reduction of scalar QED: [ S NR-sQED = S GED + d D+1 x e iv µ (m qϕ)φ D µ φ M ] iv µ (m qϕ)φd µ φ h µν D µ φ D ν φ D µ φ ( µ iq (a µ ϕτ µ ) + imm µ )φ A NR interacting theory with U m (1) U e (1) symmetries. Notice: ϕ is not in covariant derivative but appears as some kind of spacetime dependent mass or mass potential : m (x) m qϕ. ϕ = 0 is impossible to have for a dynamical GED field: Breaks boost invariance (related to w µi = 0 and sym. charges): Theory has no magnetic/electric limits.

18 Charged Galilean Point Particle Action for massive NC particle coupled to GED: S [x] = dλ 1 2 (m qϕ) h µνẋ µ ẋ ν τ µ ẋ µ + q (a µ ϕτ µ ϕm µ )ẋ µ. Flat space: EOM = NR version of the Lorentz force law: d [ (m qϕ) dt }{{} = m ẋ i ] = qẽ i + qf ij ẋ j q }{{} 2 ẋ j ẋ j i ϕ. }{{} Lorentz force-like = i m Again: ϕ = 0 is only boost covariant for non-dynamical GED. Suggestive interpretation of m here: Acts as drag or wind of the æther for the particle. Last remark: Making mass spacetime dependent is okay in NR theories unlike relativistic ones!

19 Summary

20 New Results from [1603.X] and [1603.Y] Full characterization of the coupling of M µ and relevance of the special TNC connection as minimal connection. Discovery of new kind of coupling of non-relativistic theories in sqed to mass potential m.

21 Outlook Generalizations to SUSY (null reductions should be feasible). Non-relativistic spinor QED (one also here sees the coupling to mass potential m (x) = m qϕ). NR non-abelian Yang-Mills ([ ] have some other way of doing this). See GED in non-ads holographic setups with ED in the bulk (expect GED at the boundary). Realizations of models with the coupling we see in non-relativistic sqed (anyone?).

22 Questions?

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24 Extra Slides

25 A Brief History of Space and Time Aristotle (384BC-322BC): Books on physics treating motion, causality, five elements, etc. St. Augustine of Hippo ( ): Important thoughts on time, first to state that time might have a beginning. Galileo Galilei ( ): Argued that the laws of physics are the same in all inertial systems. Isaac Newton ( ): Introduced absolute space and time in Newtonian mechanics. Woldemar Voigt ( ): First to require form invariance of physical equations. Albert Einstein ( ): Replaces Galilean relativity and Newtonian mechanics with special and general relativity.

26 Non-Unitary Galilean and Bargmann representations Galilean representation: GˆB = Bargmann representation: Bà B = 1 0 a 0 v a R a b a a a 0 v a R a b 0 a a v 2 /2 v a R a b 1 f (1). (2)

27 Transformations of Bargmann Gauge Fields δτ µ = L ξ τ µ (3) δe a µ = L ξ e a µ + λ a be b µ + Λ a τ µ (4) δv µ = L ξ v µ + e µ a Λ a (5) δe µ b = L ξ e µ b + λ a b e µ a (6) δω a µ = L ξ Ω a µ + µ Λ a + λ a bω b µ + Λ b ω a µb (7) δω ab µ = L ξ ω ab µ + µ λ ab + 2λ [a cω c b] µ, (8)

28 Curvatures of Bargmann Gauge Theory R µν (H) = 2 [µ τ ν] (9) Rµν a (P) = 2 [µ eν] a 2Ω [µ a ν] 2ω[µ a b eb ν] (10) Rµν a (B) = 2 [µ Ων] a 2ω[µ ab ν]b (11) Rµν a b (J) = 2 [µ ων] a ac b 2ω[µ ν]cb (12) R µν (M) = 2 [µ M ν] 2Ω[µ a ν]a. (13)

29 Pseudo- and special TNC Connections The natural pseudo-connection is given by ˆΩ µa v ν [ν e a µ] + v ν e σa e µb [ν e b σ] (14) ˆω µac e λ [a λe µ c] e λ [a µe λ c] e µb e σ [a eλ c] λe b σ (15) Γλ µν = ˆv λ µ τ ν hλσ ( µ h νσ + ν h µσ σ h µν ). (16) The equivalent K µν, L µν for the curvature constraint that gives the graviphotonic connection are: K σρ = 2 [σ M ρ] (17) L σµν = 2M σ [µ τ ν] 2M µ [ν τ σ] + 2M ν [σ τ µ]. (18)

30 NR Field Representations on Flat Spacetime (Bargmann) representations on fields ϕ l (t,x): ϕ l ( t,x ) = Ů ll ϕ l ( t,x ), Ů ll (ξ,λ,λ) S ll (Λ,λ) e if (t,x)m T (ξ,λ,λ) ( 1 ( ) f (t,x) Λ i 2 Λi t + δj i + λ i j )x j σ + O (3). ( Types of representations of S ll ρ SO(d) R d) : Scalar (spin-0). Spinor (spin- 1 2 ). Vector (spin-1). (Non-)relativistic mass: Galilean and Bargmann fields.

31 Conserved Bargmann Currents E can µ L [ µ ϕ l ] 0ϕ l δ µ 0 L (19) T can µi L [ µ ϕ l ] i ϕ l δ µi L (20) J can µ L i [ µ ϕ l ] (M) ll ϕ l (21) b µi can tt µi can x i J µ can + w µi. (22) j µij can x i T µj can x j T µi can + s µij (23) w µi L [ µ ϕ l ] s µij L [ µ ϕ l ] ( B i) ll ϕ l (24) ( J ij) ll ϕ l (25)

32 Gauging Global Spacetime Symmetries Again For a Bargmann or Galilean theory we introduce gauge fields τ µ, e µi, M µ, Ω µi, ω µij with coupling: [ S (1) d D x τ µ E can µ + e µi T can µi M µ J can µ + 1 ] M only Barg. 2 ω µijs µij Ω µi w µi, δ (1) τ µ = µ ɛ 0 δ (1) e µi = µ ɛ i + λ ij δµ j Λ i δµ 0 δ (1) Ω µi = µ Λ i δ (1) ω µij = µ λ ij δ (1) M µ = µ σ δµλ i i. S = S (0) + S (1) is now invariant to first order.

33 Boost Transformation of Field Strengths in Flat GED a = a v E (26) E = E (27) B = B + v E (28) Ẽ = Ẽ + v B va 1 2 v 2 E + v (v E). (29) Notice that these are different from Levi-Leblond and Le Bellac s.

34 List of Tensorial Objects in GED E ν + aτ ν (30) B µν + 2Ẽ [µ τ ν] + 2E [µ M ν] + 2aτ [µ M ν] (31) M λ E λ + a (32) B µν + 2E [µ h ν]λ M λ (33) B µν + 2E [µ v ν] (34) Ẽ ν av ν M σ (B σν + 2E [σ v ν]). (35)

35 Galilean Electrodynamics on Curved Spacetimes One can perform a null reduction of MED on a general Lorentzian manifold to get GED on a Galilean manifold: S GED = d D xe [ 1 ( ) 4 hµρ h νσ B µν + 2E [µ M ν] )(B ρσ + 2E [ρ M σ] +ˆv ρ h νσ E ν ( B ρσ + 2E [ρ M σ] ) + h νσ E ν (Ẽσ am σ ) Φh νσ E ν E σ 1 2 ( ˆv ν ˆv σ E ν E σ 2aˆv ν E ν + a 2)]. Local Galilean transformations: Field strengths B µν, E µ, Ẽ µ, a transforms in a very complicated way - not tensors! Solution: Look for combinations that are tensorial objects.

36 References Le Bellac, M. and Lévy-Leblond, Jean-Marc, Galilean electromagnetism, Il Nuovo Cimento 14 (1973) no. 2, E. S. Santos et. al., Galilean covariant Lagrangian models, J. Phys. A: Math. Gen. 37 Sep (2004) H. Künzle, Covariant Newtonian limit of Lorentz space-times, General Relativity and Gravitation 7 (1976), no. 5,

37 Sources Pictures of physicists and philosophers (checked 20/02/16): Aristotle St. Augustine of Hippo Galileo Galilei Isaac Newton Woldemar Voigt Albert Einstein