Supergravity in Quantum Mechanics

Transcription

1 Supergravity in Quantum Mechanics hep-th/ Peter van Nieuwenhuizen C.N. Yang Institute for Theoretical Physics Stony Brook University Erice Lectures, June 2017 Vienna Lectures, Jan/Feb 2017

2 Aim of these lectures Introduce the ideas of supersymmetry and supergravity without any technical details Introduce the superspace for supersymmetry and supergravity Provide a Hamiltonian approach to rigid supersymmetry which allows to compute such (anti)commutators as {Q, Q} = H; ' =[ Q, ']. For this we shall need "the Dirac formalism for constrained Hamiltonian systems", which we shall explain, to deal with Majorana spinors and Lagrange multipliers (the time components of gauge fields). For supergravity there are no Q and H but instead a BRST charge Q BRST. In the Hamiltonian BRST approach Q BRST is independent of the gauge. PvN (CNYITP) SUGRA in QM June / 33

3 History of Supersymmetry (susy) In the 1970 s, some Soviet physicists were led to supersymmetry, but by wrong motivations: Golfand + Likhtman (1971): two component (van der Waerden s dotted and undotted) spinors, explain parity violation in weak interactions: {Q,Q. } =( µ ). Pµ with Q = 1 2 (1 + 5) Volkov + Akulov (1972): neutrinos are massless because they are the Goldstone fermions of some anticommuting symmetry. Later rigid susy in general nontrivial d =4models of QFT was introduced in the West by Wess and Zumino. Q PvN (CNYITP) SUGRA in QM June / 33

4 Results right wrong heavy fermions Physical motivation good bad Einstein GR susy Dirac eqn. spontaneous emission Nordström gravity most articles Heisenberg: gravity and nongravity need each other at the quantum level. Coleman: uninterested in gravity, super-uninterested in supergravity Dirac equation: susy in QM! PvN (CNYITP) SUGRA in QM June / 33

5 Supergravity (sugra) Discovered in 1976 by Freedman, Ferrara and van Nieuwenhuizen. Useful reformulation shortly afterwards by Deser and Zumino. Usual introductions: d =4, d = 11, d = 10. Here d =1. Advantages No complicated spinor algebra ( Fierzing ) No details of QFT (only scalar fields and one-component spinors) No details of GR (no spin connection) Only zero modes Disadvantages No separate gauge action for sugra, only matter coupled to external gauge fields of supergravity (like QED without Maxwell). PvN (CNYITP) SUGRA in QM June / 33

6 Supergravity and supersymmetry (susy) Maxwell, GR, YM, Sugra Extensions with anticommuting symmetries. (Like real! complex). Square roots: sugra = p GR (Like Dirac: /@ = p 2) Legacy in math assured. Legacy in physics:? PvN (CNYITP) SUGRA in QM June / 33

7 Basic ideas Rigid susy: L = 1 2 b2b f ( p 2) f b = f and f =( p 2) b yield L =( f )2b + f ( p 2)( p 2)b =0. Equal # bosonic and fermionic states Local susy: L = 1 2 g µ 2g µ µ ( p 2) µ g µ = (µ ) and µ =( p 2 ) g µ L =( (µ ) )2g µ + µp 2 p 2 g µ =0 Building blocks: Fermion-Boson doublets. : anticommuting, spin 1/2, [ ] = 1/2. PvN (CNYITP) SUGRA in QM June / 33

8 Contents 1. Rigid supersymmetry (susy) with '(t) and (t). Algebra of rigid susy transformations. 2. Local susy = Supergravity (sugra) from the Noether method: gauge fields (t) and h(t). Closure of gauge algebra with structure functions. 3. Dirac s Hamiltonian approach for susy and sugra. Noether method for susy charge Q. Dirac brackets for Majorana spinor (and A 0 ). Superalgebra. 4. Superspace with t and, and = ' + i. Prepotential approach with H = h + i. Covariant approach with E M and constraints on supertorsions which define geometry. 5. Lagrangian BRST symmetry of quantum sugra. 6. Batalin-Fradkin-Vilkovisky s Hamiltonian BRST approach for quantum sugra: Q BRST is gauge-choice dependent. PvN (CNYITP) SUGRA in QM June / 33

9 Rigid susy in QM (I) Real '(t) and (t) (in d =1spinors have one component) L (0) = 1 2 '2 + i 2 (hermitian) (boson) fermion: ' = i ( ' real) L (0) = ' d dt (i )+ i 2 + i 2 d dt We need a derivative in : = ' Then a symmetry for = 1. L (0) = ' d dt (i )+ i 2 ( ' ) + i 2 = d i dt 2 ' d dt ( ' ) PvN (CNYITP) SUGRA in QM June / 33

10 Rigid susy in QM (II) Rigid susy algebra: ' = i, = ' [ ( 2 ), ( 1 )]' = ( 2 )[i 1 ] (1 $ 2) = i 1 ( 2 ) (1 $ 2) ( passive ) =(2i 2 1 ) d dt ' Two rigid susy transformations yield translations: rigid susy = p translations Same for (here no auxiliary fields since # bosonic fields = # fermionic fields). PvN (CNYITP) SUGRA in QM June / 33

11 Local susy from Noether method (I) For sugra, we need local (t). We get L (0) = i ' + d i dt 2 ' Introduce gauge field (t) of susy such that = + and add L N = i ' ( Noether term ) Then L (0) is canceled, but variation of ' in L N yields new terms L N = i apple d dt (i ) = i h ' ' + i i + '( ' ) PvN (CNYITP) SUGRA in QM June / 33

12 Continuation, Noether Method (II) We got L N = i ' and L N = i [ ' ' + i ] Introduce new gauge field, h(t) with h = i + Then we can cancel L N by adding L stress = h( ' ' + i ) But ambiguity: =0is the -field equation, so 0 = i into L (0) also possible. We choose the latter. Then L stress = h ' ', and h = i and 0 = i. then L N =0but again L stress = 2h ' d dt (i ) (the new 0 into L N vanishes as =0.) PvN (CNYITP) SUGRA in QM June / 33

13 Continuation, Noether Method (III) We got L stress = 2h 'i 2h 'i First term: recall L N = i ',so 0 = 2h Second term: =0is EOM, so 00 =2h ' in L (0) cancels it. But the new 00 into L N produces L N = i '(2h ' ) Canceled by 0 h =2ih into L stress. DONE! NB. Noether method can also be used to (re)derive YM and GR. PvN (CNYITP) SUGRA in QM June / 33

14 Summary of the results of the Noether Method Noether also for Einstein: Local gauge algebra: L = 1 2 ' ' + i 2 i ' h ' 2 ' = i ; = ' + i +2h ' = 2h ; h = i +2ih gauge field of GR ' = '; = ; h = ḣ h; = [ ( 2 ), ( 1 )]' = i 1 ( 2 ) (1 $ 2) =[(2i 2 1 )(1 2h)] {z } b (use the 1/2 Uniform result. Two local susys yield Einstein + susy. b : structure function. Again no auxiliary fields (again not needed). ' + i [ 2i 2 1 ] {z } b 1/2 trick") PvN (CNYITP) SUGRA in QM June / 33

15 Hamiltonian methods for rigid susy For charges Q and H we introduce p ' L = L = i 2 For Majorana spinor a problem: {, } P = 1 but {, } P =0 Solution: change the Poisson brackets. Dirac formalism: = + i 2 =0 : primary constraint H = q i p i + L = 1 2 p2 =) [, H] P =0 {, } P = i : second class constraint Dirac bracket: {A, B} D = {A, B} P {A, } P {, } 1 {, B} P P {, } D = i; {, } D = 1 2 ; {, } D = i 4 agrees with = i 2. PvN (CNYITP) SUGRA in QM June / 33

16 Equation of motion symmetries For rigid susy the action is given by L = 'p p2 But what are the transformation rules? The Hamiltonian transformation rules should not contain time derivatives, but = ' and p = d dt ' and = i 2 ( ' ). Resolution: add EOM symmetries. Example: L(f,g) for commuting f and g has EOM symmetry: Then S S S 0 S and S PvN (CNYITP) SUGRA in QM June / 33

17 Rigid susy in phase space (Hamiltonian approach) L = 'p p2 (p = '; = i 2 ) ' = i (x +(1 x)2i ) (ambiguity) EOM = = ( ' p) p = i (x +(1 S = EOM + S = i 2 = i 2 ( ' ) S = i 2 ( ' p) Results : = p ; = i 2 p p =0if x =1/2; then ' = i 2 PvN (CNYITP) SUGRA in QM June / 33

18 Canonical (Operator) approach to rigid susy Noether method gives current = charge Q and H in phase space Q = ip i 2 (was ' ) = p (if one uses Dirac brackets ) H = 1 2 p2 Q and H yield correct transformation rules of rigid susy and translations in phase space = i[, Q] D = i {, } D p = p, etc. Now we can compute the superalgebra of the charges: {Q, Q} D = 2iH; [H, Q] D =0 [H, H] D =0 Note: i~ times Dirac brackets becomes quantum (anti)commutators! PvN (CNYITP) SUGRA in QM June / 33

19 Sugra(=Local susy) in phase space The Noether method for local susy gives straightforwardly L H = 'p (1 + 2H)p2 i ( 1 2 p + ip ) ) ' = i 2 ; p =0 = i unchanged from rigid case but now = (t) 2p ; = p To obtain L L, = ; H = i L H = d dt ( p) use = + i 2 =0to replace! i 2 (since = i 2 use p =( ' i )/(1 + 2H) ) L L = H '2 + i 2 So 1 2h = 1 1+2H and = 1 1+2H i 1+2H ' PvN (CNYITP) SUGRA in QM June / 33

20 Geometry ( group manifold ) for rigid susy Geometry hg gives Lie derivatives (Q) gh gives covariant derivatives (D) g = e ith+ Q ; h = e idth+d Q Use {Q, Q} =2H; [Q, H] =0; [H, H] =0 quantum (anti)commutators gh = e (z +dz M E M )T dz M = dt, d ; T = {ih, Q} D M = In our case D = t and! D t t. ÊM 1 0 = i 1 PvN (CNYITP) SUGRA in QM June / 33

21 Recapitulation of first lecture Usual (Lagrangian) method (covariant in d > 1) L = 1 2 ' ' + i 2 i ' h ' 2 ' = i ; = ' + i +2h ' = 2h ; h = i +2ih First-order (Hamiltonian) method (not covariant in d>1) The Noether method for local susy gives straightforwardly L H = 'p (1 + 2H)p2 i ( 1 2 p + ip ) ' = i 2 ; p =0 = i 2p ; = p = ; H = i PvN (CNYITP) SUGRA in QM June / 33

22 Rigid Superspace g = e ith! e ith+ Q So coordinates t and (g unitary) ='(t)+i (t) =) Q = ' + i D anticommutes with Q: {Q, D} =0 @t Action: Z S = i 2 dtd (@ t )(D ) (dimensionless, commuting) Z S = i 2 dtd (@ t Q )(D ) t (D Q ) Z = i 2 dtd Q[@ t QD ] Z Z =0 since d f(t) =0 PvN (CNYITP) SUGRA in QM June / 33

23 Local superspace = Supergravity in Superspace Easy to find an action for sugra in superspace which reproduces the sugra action in t space. H =(1 2h)+2i : prepotential (unconstrained) Z L = i 2 d H (@ t )(D )=as before (after rescaling and ) But it is not clear how H transforms under local symmetries. Covariant approach with arbitrary supervielbeins E M ( D t E t D + E t t D M D D = E D + E t ED + Xi@ t Symmetries: super-einstein transformations =[, ]= D, D M =[, D M ] = t, where t = + i D =(D,@ t ) = + PvN (CNYITP) SUGRA in QM June / 33

24 Constraints and gauge choices Problem: t-space has 2 fields and 2 local symmetries (h,, and (t), (t)). Superspace: has4 2=8fields (E M ) and 4 local symmetries ( ). (M =( t, ) and =(t, ) and superfields have 2 components). We need 4 constraints 1 2 {D, D} = id t D =[D, ] and D t =[D t, ] There are terms with D t.solution: E M = E2 E(DX) ixẋ ix generalizes 1 {D, D} = 2 i@ dimensionally correct super-einstein covariant: ie(de) XĖ E Choose gauge which fixes 2 local symmetries such that only local susy and Einstein left: E t = ix =0 PvN (CNYITP) SUGRA in QM June / 33 t!

25 Local covariant action A real invariant (super-einstein invariant) action with the correct dimension is given by Z S = i 2 dtd (sdet E M )(D t )(D ) Z = dtd i 2E (E t )(ED ) (sdet A 0 B = det A/det B) Clearly E 2 = H is the prepotential. All other components of E M are expressed in terms of the prepotential. PvN (CNYITP) SUGRA in QM June / 33

26 Local symmetries in local superspace After the constraint: only E independent. still all After the gauge choice: only E E independent. What about? A general transformation of E M with does not respect the gauge. Only = i 2 D t allowed. Solution : t = (t) 2i (t) = (t)+ 1 2 (t) Then = D = D + t gives the results of t-space. Also E agrees. PvN (CNYITP) SUGRA in QM June / 33

27 Lagrangian BRST Symmetry of Sugra In QFT one adds a gauge fixing and ghost action: In the BRST approach L qu = L class + L fix + L ghost L fix = dh + ; L ghost =b B h + B For classical fields, BRST transformations are gauge transformations with! i and! c. For ghosts B c and B follow from nilpotency on classical fields ( B B ' =0and B B =0) For antighosts we have b = d; d = 0; = ; B B B B =0 8 Nilpotency on all fields >< L class invt.; L fix + L ghost invt. >: But BRST charge Q (L) BRST depends on L fix! PvN (CNYITP) SUGRA in QM June / 33

28 BRST symmetry for sugra in phase space Since we want to use quantum commutators, we consider sugra in phase space. Then we find BRST transformation rules, and a BRST charge from the Noether method (by using (t) ). The result is Q (H) BRST = 1 2 cp2 ( i )p + i 2 b Of course this charge is conserved, but its BRST variation does not vanish ( BRST b = d). Worse: still no momenta for H and. We now present a complete Hamiltonian framework. PvN (CNYITP) SUGRA in QM June / 33

29 The complete Hamiltonian treatment for sugra In general L qu = q I p I H D + {Q (H) BRST, g} (Dirac brackets for!) q I all fields; F g = gauge fermion ( anything ) Q (H) BRST = c + p µ (b) µ (l) 1 2 c c f p(c) ( ) First class constraints l µ = Lagrange multipliers {',' } D = f ' Sugra: H D =0 and l µ =(H, ) Q (H) BRST = 1 2 cp2 i p( i 2 )+p bp G + i p c Now {Q (H) BRST,Q(H) BRST } =0and independent of F g. L = 'p + + Ḣp H + +ċp c + ḃp b + p + + {Q (H) BRST, g} PvN (CNYITP) SUGRA in QM June / 33

30 Final results The commutators of fields with the BRST charge (times ) give all BRST transformation rules. These rules are nilpotent. They contain the classical transformation rules. The conjugate momenta of the Lagrange multipliers become the BRST auxiliary fields d and! Choosing a gauge fermion, and eliminating non-propagating fields, we recover the usual (Lagrangian) formulation of supergravity. PvN (CNYITP) SUGRA in QM June / 33

31 Extended rigid susy s N=1 d=2: ', N=2 d=1: 1 :(', 2 :(', 1 2! 1 )and( 2,F) 2 )and( 1,F), F and! 1 2 L = 1 2 '2 + i i F 2 = 1 2 p 2 and = m(f'+ i 1 2 )+g( 1 2 F'2 + i 1 2 ') p 2 and ' =ˆx +ˆpt States: e ipˆx 0 >= p>and p> PvN (CNYITP) SUGRA in QM June / 33

32 QED Dirac (1927): radiation gauge i A i =0) instantaneous Coulomb force Heisenberg and Pauli (1928): p s and q s. p(a 0 )? Fermi (1930): L = 1 4 (@ A ) 2 µ A µ phys>=0 1 HP(1930): add 2 (@ µa µ ) 2 to L. p(a 0 µ A µ. HP(1930): gauge A 0 =0 (div E - 4 ) phys>=0. Ma µ A µ phys>=0toostrong Gupta-Bleuler (1950): (@ µ A µ ) phys>=0 Dirac (1952): p(a 0 )=0primary constraint div E - 4 =0secondary constraint Both are first class. PvN (CNYITP) SUGRA in QM June / 33

33 BRST in QED Q BRST phys>=0q BRST = Z [c(dive 4 ) p(b)p(a 0 )]d 3 x Field equation p(a 0 )=@ µ A µ. Result: (div E - 4 ) + phys>=0 (@ µ A µ ) + phys>=0 Just HPFMGB! PvN (CNYITP) SUGRA in QM June / 33

34 Most importantly: Let us hope that Nature is aware of our efforts! PvN (CNYITP) SUGRA in QM June / 33