Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Gravity and the planar spin-2 Schr¨ odinger equation Eric Bergshoeff Groningen University work done in collaboration with Jan Rosseel and Paul Townsend workshop on Exceptional Field Theories, Strings and Holography Mitchell Institute, College Station April 25 2018
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Motivation
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Holography Gravity is not only used to describe the gravitational force! Christensen, Hartong, Kiritsis, Obers and Rollier (2013-2015)
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Supersymmetry supersymmetry allows to apply powerful localization techniques to exactly calculate partition functions of (non-relativistic) supersymmetric field theories Pestun (2007); Festuccia, Seiberg (2011), Pestun, Zabzine (2016)
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Condensed Matter Effective Field Theory (EFT) coupled to NC background fields serve as response functions and lead to restrictions on EFT compare to Coriolis force Luttinger (1964), Greiter, Wilczek, Witten (1989), Son (2005, 2012), Can, Laskin, Wiegmann (2014) Jensen (2014), Gromov, Abanov (2015), Gromov, Bradlyn (2017)
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Outline Newton-Cartan Geometry
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Outline Newton-Cartan Geometry Newton-Cartan Gravity
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Outline Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Outline Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Outline Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments NC Geometry in a Nutshell • Inertial frames: Galilean symmetries • Constant acceleration: Newtonian gravity/Newton potential Φ( x ) • no frame-independent formulation (needs geometry!) Riemann (1867)
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Galilei Symmetries δ t = ξ 0 but not δ t = λ i x i ! • time translations : δ x i = ξ i • space translations : i = 1 , 2 , 3 δ x i = λ i j x j • spatial rotations : δ x i = λ i t • Galilean boosts :
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments ‘Gauging’ Poincare symmetry generators gauge field parameters curvatures E µ A R µν A ( P ) space-time transl. – P A Ω µ AB Λ AB ( x µ ) R µν AB ( M ) Lorentz transf. M AB Imposing the constraint A − Ω [ µ B = 0 AB E ν ] A ( P ) ≡ 2 ∂ [ µ E ν ] R µν (‘zero torsion’) and assuming that E µ A is invertible the spin-connection field Ω µ AB becomes a dependent field : Ω µ AB Ω µ AB ( E ) →
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments ‘Gauging’ Galilei symmetry generators gauge field curvatures time translations H τ µ τ µν = ∂ [ µ τ ν ] P a e µ a R µν a ( P ) space translations G a ω µ a R µν a ( G ) Galilean boosts J ab ω µ ab R µν ab ( J ) spatial rotations Imposing Constraints does only solve for part of ω µ a , ω µ ab R µν a ( P ) = 0 :
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Absolute Time τ µν ≡ ∂ [ µ τ ν ] = 0 → τ µ = ∂ µ τ � � ∆ T = d x µ τ µ = d τ is path-independent C C
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments From Galilei to Bargmann the zero commutator [ G a , P b ] = 0 implies that a massive particle with non-zero spatial momentum P b cannot by any boost transformation G a be brought to a rest frame ⇒ [ G a , P b ] = δ ab M → extra gauge field m µ
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments The NC Transformation Rules The independent NC fields { τ µ , e µ a , m µ } transform as follows: δτ µ = 0 , δ e µ a = λ ab e µ b + λ a τ µ , δ m µ = ∂ µ σ + λ a e µ a The spin-connection fields ω µ ab and ω µ a are functions of τ µ , e µ a and m µ
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments What about the dynamics ?
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Outline Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments From General Relativity to NC gravity ‘gauging’ Poincare ⊗ U(1) = ⇒ GR plus ∂ µ M ν − ∂ ν M µ = 0 ⇓ ⇓ contraction the NC limit ‘gauging’ Bargmann = ⇒ Newton-Cartan gravity
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Contraction Poincare � � � � P A , M BC = 2 η A [ B P C ] , M AB , M CD = 4 η [ A [ C M D ] B ] 1 P 0 = 2 ω H , P a = P a , A = (0 , a ) M ab = J ab , M a 0 = ω G a Taking the limit ω → ∞ gives the Galilei algebra: � � � � P a , M b 0 = δ ab P 0 ⇒ P a , G b = 0
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Contraction Poincare ⊗ U(1) � � � � P A , M BC = 2 η A [ B P C ] , M AB , M CD = 4 η [ A [ C M D ] B ] Z plus 1 Z = 1 P 0 = 2 ω H + ω Z , 2 ω H − ω Z , A = (0 , a ) P a = P a , M ab = J ab , M a 0 = ω G a Taking the limit ω → ∞ gives the Bargmann algebra including Z: � � � � P a , M b 0 = δ ab P 0 ⇒ P a , G b = δ ab Z
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments The NC Limit I Dautcourt (1964); Rosseel, Zojer + E.B. (2015) express relativistic fields { E µ A , M µ } in terms of non-relativistic STEP I: fields { τ µ , e µ a , m µ } E µ 0 = ω τ µ + 1 M µ = ω τ µ − 1 E µ a = e µ a 2 ω m µ , 2 ω m µ , 1 constraint : ∂ [ µ τ ν ] = 2 ω 2 ∂ [ µ m ν ]
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments The NC Limit II STEP II: substitute the expressions into the transformation rules and the e.o.m. and take the limit ω → ∞ ⇒ • the NC transformation rules are obtained and agree with the gauging procedure • the NC equations of motion are obtained Note: the standard textbook limit gives Newton gravity
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments The NC Equations of Motion The NC equations of motion are given by ´ Elie Cartan 1923 c ( G ) c ( G ) = 0 τ µ e ν c R µν = R 0 c 1 ca ( J ) ca ( J ) = 0 τ µ e ν c R µν = R 0 c a e µ ( a e ν c R µν cb ) ( J ) R ( accb ) ( J ) = 0 = ( ab ) • there is no known action that gives rise to these equations of motion • after gauge-fixing τ µ = δ µ, 0 , e µ a = δ µ a and m 0 = Φ the 4D NC e.o.m. reduce to △ Φ = 0
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments what about non-relativistic matter?
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Outline Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Motivation special feature FQH Effect: existence of a gapped collective non-rel. parity non-invariant helicity-2 excitation, known as the GMP mode Girvin, MacDonald and Platzman (1985) recent proposal for a non-relativistic spatially covariant bimetric EFT describing non-linear dynamics of this massive spin-2 GMP mode Haldane (2011); Gromov and Son (2017) in a linearized approximation around a flat background this gives rise to a single spin-2 Planar Schr¨ odinger Equation Ψ + � 2 i � ˙ 2 m ∇ 2 Ψ = 0
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments Key Question Rosseel, Townsend + E.B., to appear in PRL has this single helicity 2 Planar Schr¨ odinger Equation a (massive) gravity origin?
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments The ‘force limit’ of spin 0 1 � 2 � mc c 2 ¨ Φ − ∇ 2 Φ + Φ = 0 � Take the non-relativistic limit c → ∞ keeping λ = � / mc fixed → ∇ 2 Φ = 1 λ 2 Φ no massive spin 0 particle !
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments The ‘particle limit’ of complex spin 0 1 � 2 � mc c 2 ¨ Φ − ∇ 2 Φ + Φ = 0 � To avoid infinities we redefine Φ = e − i � ( mc 2 ) t Ψ so that the Klein-Gordon equation becomes Ψ − � 2 1 i � d � 2 � Ψ − i � ˙ 2 m ∇ 2 Ψ = 0 − 2 mc 2 dt and the c → ∞ limit yields the Schr¨ odinger equation Ψ + � 2 i � ˙ 2 m ∇ 2 Ψ = 0
Newton-Cartan Geometry Newton-Cartan Gravity Non-Relativistic Matter Comments General Feature one complex massive helicity mode ⇔ one Schr¨ odinger Equation
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