SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

More documents

Recommendations

Info

372 P. <strong>van</strong> Nieuwenhuizen. SupergravityIn what follows we will show that(I) fixing the gauge in the path integral one finds as a generalization of the Faddeev—Popovdeterminant a superdeterminant;(ii) exponentiating this superdeterminant by means of a super Gaussian integral one finds thegeneralized ghost actions.Consider a path-integral with bosonic and fermionic symmetries. We want to multiply by unitywriting as usual1 = Jd~”d~ 1,0(Fa foe) o(B 1, — b1,)J (5)where ~(~)are the bosonic (fermionic) gauge parameters, B(F) the gauge choices and b(f) ordinary(Grassmann) variables. We must first define the Dirac delta function for fermionic arguments. SincefdOO = 1, it follows that 0(O)= 0. Thus 0(F,. —f,.) is equal to (F,. —f,.) for fermionic F,. —f,,. Making achange of integration variables (~,~)-÷ (B, F), it follows that J is the Jacobian for the change ofintegration variables (B, F)—~~ (~,~j).Theorem: The Jacobian for x” —~ x’(x’, 0’),9”’ —* 0”(x’, 0’) is given by the superdeterminant1,/3x’)] (6)—det[~x”/3x’— (19f/o0”) det(t90”’/t991,) (30’/39)~”1,(89’where it does not matter whether one uses left or right derivatives except that lJx”/90” is a rightderivative. (In all other cases we use left derivatives.)We will begin by considering linear transformation(x’\(A B\fx 7~o’i ~C DROand derive that J = sdet M In this case it is clear that 3x”/30” is a right derivative. Consider first thecase of linear changes of anticommuting variables 0’ (i = 1, N). FromJ d9~ .dONOSJ 61=1=JdOhI...dOlNO7~’~=Jdo1...deNJo~N...o~1(8)with 0” = D’,O’ it follows that J = det D’, just the inverse oL what one has for bosonic variables. Forgeneral linear transformations involving commuting x’ and anticommuting 90 one has/x’\ fA B\ fx\ /1 BD1\ fA—BD’C 0\ / I O\ fx 9~o’) = ~C D) ~e) = ~o i I ~ 0 D) ~D1C i) ~o ()The first and last transformations are shifts of integration variables 0” and x’ respectively, and since theintegrals are translationally invariant these Jacobians equal unity. The second change of integration
P. <strong>van</strong> Nieuwenhuizen, Supergravity 373variables is diagonal in x and 0, and one finds indeed that for linear changes of integration variables thesuper Jacobian is equal to the superdeterminant.For nonlinear changes of integration variables, the super Jacobian is given by the result previouslygiven, or by the equivalent resultJ = det(3x”/ôx’) det’(3O”’/aO1, — 3O”’/9x~(Dx’/ax)1~3x”/301,) (10)and this result is derived in the next subsection.Finally we discuss how to exponentiate the superdeterminant. We claimsdetM= JdCexpiciJvieiC~, dC=fldC~dC~ (11)where C~denote commuting as well as anticommuting ghosts, namelyC~M,,C’2= C~A,,,.C2”+ C~B,,,.C2”+ C~CO,.C~ + C~’DO1,C~ (12)and C~’are anticommuting while C’ are commuting. The proof is based on translational invariance0= (—~(0M‘)“ fdC (exp(iC1MC2))= (—Y(0M 1)0 J dC[iM,1— M,kC~C~M,,]exp(iC1MC2)= fdC[istr(OM 1M)— C~OMIk C~]exp(1C 1MC2) (13)since interchanging [(8M 1)MC 2]’ and (C,My cancels the factor (—)‘ and yields an extra factor (—1)since ghosts have different statistics.WithI = fdC exp(iC1MC2) and str(0M’M) = —str(OMM~) (14)one finds thus thatOlnI=OlnsdetM (15)which is indeed what we intended to prove.One often rewrites the ghost action as1. (16)..~‘(ghost)= C*IM,1CIndeed, the Feynman rules are the same, but since (for example in Yang—Mills theory) M is not
Page 3 and 4:
In memoriam Joel ScherkJoel Scherk
Page 6 and 7:
194 P. van Nieuwenhuizen, Supergrav
Page 8 and 9:
Page 10 and 11:
Page 12 and 13:
200 P. van Nieuwenhuizen. Supergrav
Page 14 and 15:
202 P. tan Nieuwenhuizen, Supergrav
Page 16 and 17:
Page 18 and 19:
Page 20 and 21:
208 P. van Nieuwenhuizen. Supergrar
Page 22 and 23:
Page 24 and 25:
Page 26 and 27:
Page 28 and 29:
Page 30 and 31:
Page 32 and 33:
220 P. ran Nieuwenhuizen. Supergrar
Page 34 and 35:
Page 36 and 37:
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
Page 46 and 47:
Page 48 and 49:
Page 50 and 51:
Page 52 and 53:
Page 54 and 55:
Page 56 and 57:
Page 58 and 59:
Page 60 and 61:
Page 62 and 63:
250 P. ran Nieuwenhuizen, Supergrav
Page 64 and 65:
Page 66 and 67:
Page 68 and 69:
Page 70 and 71:
Page 72 and 73:
Page 74 and 75:
Page 76 and 77:
Page 78 and 79:
Page 80 and 81:
268 P. Lan Nieuwenhui:en. SupergraL
Page 82 and 83:
Page 84 and 85:
Page 86 and 87:
Page 88 and 89:
Page 90 and 91:
278 P. t’an Nieuwenhuizen, Superg
Page 92 and 93:
Page 94 and 95:
Page 96 and 97:
Page 98 and 99:
Page 100 and 101:
Page 102 and 103:
Page 104 and 105:
Page 106 and 107:
Page 108 and 109:
Page 110 and 111:
Page 112 and 113:
Page 114 and 115:
Page 116 and 117:
Page 118 and 119:
Page 120 and 121:
Page 122 and 123:
Page 124 and 125:
Page 126 and 127:
(A V,.”)00 = ~ s*a = (D,. + ~ A,.
Page 128 and 129:
Page 130 and 131:
Page 132 and 133:
Page 134 and 135: 322 P. van Nieuwenhuizen, Supergrav
Page 136 and 137: 324 P. van Nieuwenhuizen. Supergrav
Page 140 and 141: 328 P. van Nicuwenhuizen, Supergrav
Page 142 and 143: 330 P. van Nieuwenhuizen Supergravi
Page 166 and 167: 354 P. van Nieuwenhuizen Supergravi
Page 204 and 205: 392 P. van Nieuwenhui’zen, Superg
Page 210: 398 P. van Nieuwenhuizen, Supergrav
show all

SUPERGRAVITY P. van NIEUWENHUIZEN To Joel Scherk 0370 ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?