Københavns Universitet University of Copenhagen Logo: RGB 144/26/30 Logotype: RGB 102/102/102 Principopsætning på digitalmedier Placement of logo on digital media Principopsætning til andre formål Use of logo for other purposes Vandret variant med logo til venstre Vandret variant med logo til højre Horizontal version with logo to the right Horizontal version with logo to the left Lodret variant med navnetræk placeret under logostregen. Vertical version with logotype under the logo line 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 .