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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) Outline
Tekst starter uden Introduction
Non-Relativistic Fields and NC
Galilean Electrodynamics
Summary
og ”Enhedens Tekst starter uden
Introduction
og ”Enhedens Brief ReviewFigures of (Torsional) Newton-Cartan 19. februar 2016 17:53
Tekst starter uden
µν ij µ ν • Newton-Cartan geometry with τµ, h = δ ei ej (see for example [1311.4794], [1409.5555], [1504.07461]). i i λ • Covariant derivatives contains gauge fields Ωρ , ωρ j , Γρµ: λ ∇ρτµ = ∂ρτµ − Γρµτλ = 0 i i λ i i i j ∇ρeµ = ∂ρeµ − Γρµeλ − Ωρ τµ − ωρ j eµ = 0. • In TNC there is no “Levi-Civita connection” [1412.8212]. og ”Enhedens
Defense side 1 • There exists a unique connection linear in Mµ transforming µ under infinitesimal Galilean boosts λi and GCTs ξ as i δMµ = LξMµ + eµλi . • The affine connection of this special TNC connection: 1 ˚Γλ = −vˆλ∂ τ + hλσ ∂ h + ∂ h − ∂ h µν µ ν 2 µ νσ ν µσ σ µν µ µ µλ vˆ ≡ v − h Mλ , hµν ≡ hµν − 2τ(µMν) . λ λ • This is torsionful connection with Tµν = −2vˆ ∂[µτν]. • It is not clear that this a particularly interesting or useful connection: Noether procedure clears this up.
Special TNC connection
• What is the connection that is most analog to the Levi-Civita
Tekst starter uden connection?
og ”Enhedens Special TNC connection
• What is the connection that is most analog to the Levi-Civita
Tekst starter uden connection?
• There exists a unique connection linear in Mµ transforming µ under infinitesimal Galilean boosts λi and GCTs ξ as i δMµ = LξMµ + eµλi . • The affine connection of this special TNC connection: 1 ˚Γλ = −vˆλ∂ τ + hλσ ∂ h + ∂ h − ∂ h µν µ ν 2 µ νσ ν µσ σ µν µ µ µλ vˆ ≡ v − h Mλ , hµν ≡ hµν − 2τ(µMν) . λ λ • This is torsionful connection with Tµν = −2vˆ ∂[µτν]. • It is not clear that this a particularly interesting or useful connection: Noether procedure clears this up. og ”Enhedens Tekst starter uden
Non-Relativistic Fields and NC
og ”Enhedens Conserved Noether Currents of an Action S(0) = R dDx L(ϕ,∂ϕ)
• A NR (Bargmann) field with mass M transforms as Tekst starter uden 0 0 0 if (t,x)M d ϕ` t ,x = e S``ϕ` (t,x) , S`` ∈ rep SO(d) n R . • Noether’s theorem and conservation laws for action S(0): µ µi µ Ecan , Tcan , Jcan , energy momentum mass, only present for M6=0 µi µi i µ µi µij [i |µ|j] µij bcan = tTcan − x Jcan + w , jcan = 2x Tcan + s . boost ang. mom. • The non-conserved spin- and lift-currents sµij , w µi . • Conservation laws for boost and rotation currents: µi 0i i µi ∂µbcan = 0 ⇒ Tcan = Jcan − ∂µw µij [ij] µij ∂µjcan = 0 ⇒ 2Tcan = −∂µs . • Massless NR theories are very different from rel. ones: 0i 0i µi og ”Enhedens Symmetry charge densities Tcan = bcan = 0 when w = 0 . Improvements of Currents
• Currents are only defined up to a total derivative, i.e. they µi µi ρµi Tekst starter uden can be improved as Timp = Tcan + ∂ρB . • Current simplifications and choices leading to maximal simplification, important result: µij [i |µ|j] ij ji jimp = 2x Timp ⇒ Timp = Timp µi µi i µ 0i i M 6= 0 : bimp = tTimp − x Jimp ⇒ Timp = Jimp
µi µi µi 0i µi M = 0 : bimp = tTimp + ψ ⇒ Timp = −∂µψ h i 1 ψµi ≡ δµ w 0i − δµ w [ij] + s0ij . 0 j 2 • ψµi is non-conserved but is related to improvements µ ρµ µ Φ ≡ ∂ρC of the mass current J : Φµ ≡ − ∂ ψ0j + ∂ ψij δµ + ∂ ψ0j δµ . og ”Enhedens 0 i j j 0 Gauging Global Spacetime Symmetries
• Notice: The variation of the action wrt. local parameters:
Tekst starter uden Z 0 D (0)µN δS [ϕ] = − d x ∂µξN (x)J . M • Noether procedure: Making global symmetry local by introducing gauge fields:
S [ϕ,A] = S(0) [ϕ] + S(1) [ϕ,A] + S(2) [ϕ,A] + ... universal! Z (1) D (0)µN (1) S [ϕ,A] = d x AµN J , δ AµN ≡ ∂µξN . M • For a Bargmann or Galilean theory we introduce gauge fields
τ µ, eµi , Mµ, Ωµi , ωµij with coupling: Z " # (1) D µ µi µ 1 µij µi S ≡ d x τ µEcan + eµi Tcan − MµJcan + ωµij s − Ωµi w , M only Barg. 2 og ”Enhedens Gauging Global Bargmann Spacetime Symmetries
• Express canonical currents in terms of maximally simplifying
Tekst starter uden improvements and do integration by parts: Z (1) D µ 0i 1 ij µ S = d x τ µEcan + e0i Timp + sij Timp − MµJimp M 2 1 + C sµij − C w µi , v ≡ −e , s ≡ 2e , 2 µij µi i 0i ij (ij)
˚ ˚ ˚ C µi ≡ Ωµi − Ωµi , C µij ≡ ωµij − ˚ωµij 1 Ω˚ ≡ −δk ∂ s + ∂ v − 2δ0 ∂ M − δk ∂ M µi µ 2 0 ki (k i) µ [0 i] µ [k i] ˚ k 0 ωµij ≡ δµ ∂[i sj]k − ∂k e[ij] − δµ ∂0e[ij] + ∂[i v j] − ∂[i Mj] .
• Extracting eµi ,Mµ-dependent connection from improvements. ˚ ˚ • Contortions C µi , C µij and minimal coupling. og ”Enhedens Gauging Global Galilean Spacetime Symmetries
• No coupling to Mµ: Inserting the maximally simplified
Tekst starter uden improved currents is now a different story: Z (1) D µ 0i 1 ij 1 µij µi S = d x τ µEcan +e0i Timp + sij Timp + C µij s −C µi w , M 2 2 ˆ C µi ≡ Ωµi − Ωµi , C µij ≡ ωµij − ωˆµij 1 Ωˆ ≡ −δk ∂ s + ∂ v µi µ 2 0 ki (k i) ˆ k 0 ωµij ≡ δµ ∂[i sj]k − ∂k e[ij] − δµ ∂0e[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 Φµ from ψµi . µ • Add coupling MµΦ . og ”Enhedens Reproducing Newton-Cartan Geometry
• Identifying gauge fields and vielbeins: Linearization
Tekst starter uden 0 i i i τµ = δµ + τ µ + O (2) , eµ = δµ + eµ + O (2) . ˆ • Identifying the pseudo-connection Ωµi , ωˆµij in Galilean theories: No “Levi-Civita-like” connection. ˚ • Further Ωµi , ˚ωµij is the special TNC connection and hence we conclude the minimal TNC connection! ˚ • For the special TNC connection Ωµi , ˚ωµij coupling is always: µ µ Mµ (Jcan + Φ ) . • However there are other “α ∈ R” connections involving just Mµ not predicted by Noether procedure [1504.07461]: 1 Γλ (α) = −vˆλ∂ τ + hλσ (∂ H + ∂ H − ∂ H ) µν µ ν 2 µ νσ ν µσ σ µν ˜ ˜ µ µν og ”Enhedens Hµν (α) ≡ hµν + αΦτµτν , Φ ≡ −v Mµ + ½h MµMν . Tekst starter uden
Galilean Electrodynamics
og ”Enhedens Galilean Electrodynamics on Flat Spacetime
• Maxwellian electrodynamics is a relativistic U(1) gauge theory.
Tekst starter uden • There are two non-relativistic limits at the level of the Maxwell EOMs [Le Bellac & Lévy-Leblond 1973]: Z " # F = d3x ρE + J × B , electric effects magnetic effects • Null reduction (dimensional reduction along null direction) of MED: Another theory of non-relativistic “electrodynamics”: 1 1 L = − B · B + E · E˜ + a2 − ρϕ˜ + J · A − ρϕ.˜ 2 2 • ED with four field strengths B, E, E˜, a expressed in terms of three potentials ϕ,˜ A, ϕ (see [Santos et. al. 2005]): i i j k ˜ i i i B ≡ jk ∂ A E ≡ −∂ ϕ˜ − ∂t A i i E ≡ −∂ ϕ a ≡ −∂t ϕ. og ”Enhedens Galilean Electrodynamics on Curved Spacetimes
Tekst starter uden • 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 µ ν µ µν ∇µZ = −2∂[µτν]vˆ Z E ≡ h Eν ˚ µ µν [µ ν] λµ λµ [λ µ] ∇µE = −∂[µτν] W − 2vˆ E W ≡ B + 2E v ˚ µρ µ νρ ρ µν ρ ˜ ρ ρ σρ ∇µW = −∂[µτν] (2vˆ W +v ˆ W ) Z ≡ E − v a − MσW .
• First to study GED on general TNC (as far as we know): compare to literature [Künzle 1976], [1512.03799]. • Sources can be added.
og ”Enhedens Non-Relativistic Scalar QED
• Perform a null reduction of scalar QED: Z Tekst starter uden D+1 µ ∗ SNR-sQED = SGED + d x e iv (m − qϕ)φ Dµφ M µ ∗ µν ∗ −iv (m − qϕ)φDµφ − h Dµφ Dνφ
Dµφ ≡ (∂µ − iq (aµ − ϕτ˜ µ) + imMµ)φ
• A NR interacting theory with Um (1) × Ue (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. og ”Enhedens Charged Galilean Point Particle
• Action for massive NC particle coupled to GED:
Tekst starter uden Z µ ν 1 hµνx˙ x˙ µ S [x] = dλ (m − qϕ) µ + q (aµ − ϕτ˜ µ − ϕMµ)x ˙ . 2 τµx˙ • Flat space: EOM = NR version of the Lorentz force law:
d h i j q j (m − qϕ)x˙i = qE˜i + qFij x˙ − x˙ x˙j ∂i ϕ . dt | {z } | {z } 2 |{z} =m ˜ 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! og ”Enhedens Tekst starter uden
Summary
og ”Enhedens New Results from [1603.X] and [1603.Y]
Tekst starter uden
• Full characterization of the coupling of Mµ 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˜ .
og ”Enhedens Outlook
Tekst starter uden • 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 ([1512.08375] 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?).
og ”Enhedens Tekst starter uden
Questions?
og ”Enhedens
Tekst starter uden
Extra Slides
og ”Enhedens A Brief History of Space and Time
• Aristotle (384BC-322BC): Books on physics treating motion, Tekst starter uden causality, five elements, etc. • St. Augustine of Hippo (354-430): Important thoughts on time, first to state that time might have a beginning. • Galileo Galilei (1564-1642): Argued that the laws of physics are the same in all inertial systems. • Isaac Newton (1643-1727): Introduced absolute space and time in Newtonian mechanics. • Woldemar Voigt (1850-1919): First to require form invariance of physical equations. • Albert Einstein (1879-1955): Replaces Galilean relativity and Newtonian mechanics with special and general relativity. og ”Enhedens Non-Unitary Galilean and Bargmann representations
Galilean representation: Tekst starter uden
1 0 a0 Aˆ G = v a Ra aa (1) Bˆ b 0 0 1
Bargmann representation:
1 0 0 a0 a a a A˜ v R 0 a B = b . (2) B˜ −v2/2 −v Ra 1 f a b 0 0 0 1
og ”Enhedens Transformations of Bargmann Gauge Fields
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δτµ = Lξτµ (3) a a a b a δeµ = Lξeµ + λ beµ + Λ τµ (4) µ µ µ a δv = Lξv + ea Λ (5) µ µ a µ δeb = Lξeb + λb ea (6) a a a a b b a δΩµ = LξΩµ + ∂µΛ + λ bΩµ + Λ ωµb (7) ab ab ab [a |c|b] δωµ = Lξωµ + ∂µλ + 2λ c ωµ , (8)
og ”Enhedens Curvatures of Bargmann Gauge Theory
Tekst starter uden
Rµν (H) = 2∂[µτν] (9) a a a a b Rµν (P) = 2∂[µeν] − 2Ω[µ τν] − 2ω[µ |b|eν] (10) a a ab Rµν (B) = 2∂[µΩν] − 2ω[µ Ων]b (11) a a ac Rµν b (J) = 2∂[µων] b − 2ω[µ ων]cb (12) a Rµν (M) = 2∂[µMν] − 2Ω[µ eν]a . (13)
og ”Enhedens Pseudo- and special TNC Connections
The natural pseudo-connection is given by
Tekst starter uden
ˆ ν a ν σa b Ωµa ≡ v ∂[νeµ] + v e eµb∂[νeσ] (14) λ λ σ λ b ωˆµac ≡ e [a|∂λeµ|c] − e [a|∂µeλ|c] − eµbe [ae c]∂λeσ (15)
1 ˚Γλ = −vˆλ∂ τ + hλσ ∂ h + ∂ h − ∂ h . (16) µν µ ν 2 µ νσ ν µσ σ µν
The equivalent Kµν, Lµν for the curvature constraint that gives the graviphotonic connection are:
Kσρ = 2∂[σMρ] (17)
Lσµν = 2Mσ∂[µτν] − 2Mµ∂[ντσ] + 2Mν∂[στµ] . (18) og ”Enhedens NR Field Representations on Flat Spacetime
• (Bargmann) representations on fields ϕ` (t,x): Tekst starter uden 0 0 0 ˚ 0 0 ϕ` t ,x = U``ϕ` t ,x ,
˚ −if (t,x)M U`` (ξ,Λ,λ) ≡ S`` (Λ,λ) × e × T (ξ,Λ,λ) 1 f (t,x) ≡ Λ Λi t + δi + λi x j − σ + O (3) . i 2 j j
d • Types of representations of S`` ∈ ρ SO(d) n R : • Scalar (spin-0). 1 • Spinor (spin- 2 ). • Vector (spin-1). • (Non-)relativistic mass: Galilean and Bargmann fields. og ”Enhedens Conserved Bargmann Currents
∂L Tekst starter uden µ µ Ecan ≡ ∂0ϕ` − δ0 L (19) ∂ [∂µϕ`] µi ∂L i µi Tcan ≡ ∂ ϕ` − δ L (20) ∂ [∂µϕ`] µ ∂L Jcan ≡ −i (M)`` ϕ` (21) ∂ [∂µϕ`] µi µi i µ µi bcan ≡ tTcan − x Jcan + w . (22) µij i µj j µi µij jcan ≡ x Tcan − x Tcan + s (23)
µi ∂L i w ≡ − B ϕ` (24) ∂ [∂µϕ`] `` µij ∂L ij s ≡ − J ϕ` (25) ∂ [∂µϕ`] `` og ”Enhedens Gauging Global Spacetime Symmetries Again
• For a Bargmann or Galilean theory we introduce gauge fields
Tekst starter uden τ µ, eµi , Mµ, Ωµi , ωµij with coupling:
Z " # (1) D µ µi µ 1 µij µi S ≡ d x τ µEcan + eµi Tcan − MµJcan + ωµij s − Ωµi w , M only Barg. 2
(1) δ τ µ = ∂µ0 (1) j 0 δ eµi = ∂µi + λij δµ − Λi δµ (1) δ Ωµi = −∂µΛi (1) δ ωµij = ∂µλij (1) i δ Mµ = ∂µσ − δµΛi .
• S = S(0) + S(1) is now invariant to first order. og ”Enhedens Boost Transformation of Field Strengths in Flat GED
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a0 = a − v · E (26) E 0 = E (27) B0 = B + v × E (28) 0 1 E˜ = E˜ + v × B − va − v2E + v (v · E) . (29) 2
Notice that these are different from Levi-Leblond and Le Bellac’s.
og ”Enhedens List of Tensorial Objects in GED
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Eν + aτν (30) ˜ Bµν + 2E[µτν] + 2E[µMν] + 2aτ[µMν] (31) λ MλE + a (32) µν [µ ν]λ B + 2E h Mλ (33) Bµν + 2E [µv ν] (34) ν ν σν [σ ν] E˜ − av − Mσ B + 2E v . (35)
og ”Enhedens Galilean Electrodynamics on Curved Spacetimes
Tekst starter uden • One can perform a null reduction of MED on a general Lorentzian manifold to get GED on a Galilean manifold:
Z 1 S = dDxe − hµρhνσ B + 2E M B + 2E M GED 4 µν [µ ν] ρσ [ρ σ] ρ νσ νσ ˜ +ˆv h Eν Bρσ + 2E[ρMσ] + h Eν Eσ − aMσ 1 −Φ˜hνσE E − vˆνvˆσE E − 2avˆνE + a2 . ν σ 2 ν σ ν
• Local Galilean transformations: Field strengths Bµν, Eµ, E˜µ, a transforms in a very complicated way - not tensors! • Solution: Look for combinations that are tensorial objects.
og ”Enhedens References
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Le Bellac, M. and Lévy-Leblond, Jean-Marc, “Galilean electromagnetism”, Il Nuovo Cimento 14 (1973) no. 2, 217-234. E. S. Santos et. al., “Galilean covariant Lagrangian models”, J. Phys. A: Math. Gen. 37 Sep (2004) 9771–9789. H. Künzle, “Covariant Newtonian limit of Lorentz space-times”, General Relativity and Gravitation 7 (1976), no. 5, 445–457.
og ”Enhedens Sources
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• Pictures of physicists and philosophers (checked 20/02/16): • Aristotle • St. Augustine of Hippo • Galileo Galilei • Isaac Newton • Woldemar Voigt • Albert Einstein
og ”Enhedens