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Functional Integration: Action and Symmetries Functional Integration: Action and Symmetries P. Cartier and C. DeWitt-Morette 3 Acknowledgements Throughout the years, several institutions and their directors have pro- vided the support necessary for the research and completion of this book. From the inception of our collaboration in the late seventies to the con- clusion of this book, the Institut des Hautes Etudes Scientifiques (IHES) at Bures-sur-Yvette has provided “La paix n´ecessaire `aun travail intellectuel intense et la stimulation d’un auditoire d’´elite” . We have received much † help from the Director, J.P. Bourguignon, and the intelligent and always helpful supportive staff of the IHES. Thanks to a grant from the Lounsbery Foundation in 2003 C. DeW. has spent three months at the IHES. Among several institutions which have given us block of uninterrupted time, the Mathematical Institute of the University of Warwick played a special role thanks to K. David Elworthy and his mentoring of one of us (C. DeW.). In the Fall 2002, one of us (C.DeW.) was privileged to teach a course at the Sharif University of Technology (Tehran), jointly with Neda Sadooghi. C.DeW. created the course from the first draft of this book; the quality, the motivation, and the contributions of the students (16 men, 14 women) made teaching this course the experience that we all dream of. The Department of Physics, and the Center for Relativity of the University of Texas at Austin, have been home to one of us, and a welcoming retreat to the other. Thanks to Alfred Schild, founder and director of the Center for Relativity, one of us (C.DeW.) resumed a full scientific career after sixteen years cramped by alleged nepotism rules. This book has been so long on the drawing board that many friends have contributed to its preparation. One of them, Alex Wurm, has helped C. DeW. in all aspects of the preparation from critical comments to typing the final version. C´ecile thanks her graduate students My career began on October 1, 1944. My gratitude encompasses many teachers and colleagues. The list would be an exercise in name-dropping. For this book I wish to bring forth the names of those who have been my graduate students. Working with graduate students has been the most rewarding experience of my professional life. In a few years the relationship evolves from guiding a student to being guided by a promising young colleague. Dissertations often begin with a challenging statement. When completed, An expression of L. Rosenfeld. † 4 a good dissertation is a wonderful document, understandable, carefully crafted, well referenced, presenting new results in a broad context. I am proud and humble to thank: Michael G.G. Laidlaw (Ph.D. 1971, UNC Chapel Hill) Quantum Mechanics in Multiply Connected Spaces. Maurice M. Mizrahi (Ph.D. 1975, UT Austin) An Investigation of the Feyn- man Path Integral Formulation of Quantum Mechanics. Bruce L. Nelson (Ph.D. 1978, UT Austin) Relativity, Topology, and Path Integration. Benny Sheeks (Ph.D. 1979, UT Austin) Some Applications of Path Integra- tion Using Prodistributions. Theodore A. Jacobson, (Ph.D. 1983, UT Austin) Spinor Chain Path Integral for the Dirac Electron. Tian Rong Zhang, (Ph.D. 1985, UT Austin) Path Integral Formulation of Scattering Theory With Application to Scattering by Black Holes. Alice Mae Young (Ph.D. 1985, UT Austin) Change of Variable in the Path Integral Using Stochastic Calculus. Charles Rogers Doering (Ph.D. 1985, UT Austin) Functional Stochastic Dif- ferential Equations: Mathematical Theory of Nonlinear Parabolic Systems with Applications in Field Theory and Statistical Mechanics. Stephen Low (Ph.D. 1985, UT Austin) Path Integration on Spacetimes with Symmetry. John La Chapelle (Ph.D. 1995, UT Austin) Functional Integration on Sym- plectic Manifolds. Clemens S. Utzny (Master 1995, UT Austin) Application of a New Approach to Functional Integration to Mesoscopic Systems. Alexander Wurm (Master 1995, UT Austin) Angular Momentum-to-Angular Momentum Transition in the DeWitt/Cartier Path Integral Formalism. Xiao-Rong Wu-Morrow (Ph.D. 1996, UT Austin) Topological Invariants and Green’s Functions on a Lattice. Alexander Wurm (Diplomarbeit, 1997, Julius-Maximilians-Universit¨at, W¨urz- burg), The Cartier/DeWitt Path Integral Formalism and its Extension to Fixed Energy Green’s Functions. David Collins (Ph.D. 1997, UT Austin) Two-State Quantum Systems Inter- acting with their Environments: A Functional Integral Approach. 5 Christian Saemann (Master 2001, UT Austin) A new representation of cre- ation/annihilation operators for supersymmetric systems. Matthias Ihl (Master 2001, UT Austin) The Bose/Fermi oscillators in a new supersymmetric representation. Gustav Markus Berg (Ph.D. 2001, UT Austin) Geometry, Renormalization, and Supersymmetry. Alexander Wurm (Ph.D. 2002, UT Austin) Renormalization Group Appli- cations in Area-Preserving Nontwist Maps and Relativistic Quantum Field Theory. Marie E. Bell (Master 2002, UT Austin) Introduction to Supersymmetry. 6 Symbols A := B A is defined by B A = B both sides are equal only after they are R integrated B > A B is inside the light cone of A − d×l = dl/l multiplicative differential ∂×/∂l = l∂/∂l multiplicative derivative D D R , RD are dual of each other; R is a space of contra- variant vectors, RD is a space of covariant vectors D D R × space of D by D matrices X, X′ X′ is dual to X x ,x dual product of x X and x X h ′ i ∈ ′ ∈ ′ (x y) scalar product of x, y X, assuming a metric | ∈ MD, g D-dimensional riemannian space with metric g T M tangent bundle over M ¡ ¢ T ∗M contangent bundle over M Lie derivative in the X-direction LX U 2D(S) Space of critical points of the action functional S (Ch 4) MD Space of paths with values in MD, satisfying Pµ,ν µ initial conditions and ν final conditions ¡ ¢ U := U 2D(S) MD arena for WKB µ,ν ∩ Pµ,ν } ¡ ¢ ~ = h/2π Planck’s constant 2 1 [h]= ML T − physical (engineering) dimension of h ω = 2πν ν frequency, ω pulsation t = i~β = i~k T (1.70) B − − B τ = it (1.100) Superanalysis (Ch 9) A˜ parity of A 0, 1 ∈ { } AB = ( 1)A˜B˜ BA graded commutator or [A, B] (9.5) − A B = ( 1)A˜B˜ B A graded anticommutator A, B (9.6) ∧ − − ∧ { } ξµξα = ξαξµ Grassmann generators (9.11) − z = u + v supernumber, u even C , v odd C (9.12) ∈ c ∈ a R real elements of (9.16) c ⊂ Cc Cc 7 R C real elements of (9.16) a ⊂ a Ca z = zB + zS supernumber zB body, zS soul (9.12) xA = (xa, ξα) Rn ν superpoints ∈ | 8 Conventions Fourier transforms D D ( f) x′ := d x exp 2πi x,x′ f(x) x R ,x′ RD F RD − h i ∈ ∈ Z For Grassmann¡ ¢ variables ¡ ¢ ( f) (κ) := dξ exp( 2πiκξ) f(ξ) F − Z In both cases δ, f = f(0) i.e. δ(ξ)= c 1ξ h i − 2 1 δ, f = f i.e. c = (2πi)− hF i 2 1 dξ ξ = c, here c = (2πi)− Z 9 Formulary (giving a context to symbols) Wiener integral • τb E exp dτ V (q(τ) (1.1) − · µ Zτa ¶¸ Peierls bracket • A˜B˜ (A, B) := D−B ( 1) D−B (1.9) A − − A Schwinger variational principle • δ A B = i A δS/~ B (1.11) h | i h | | i Quantum partition function • βHˆ Z(β) = Tr e− (1.71) ³ ´ Schr¨odinger equation • 1 2 1 i∂tψ(x,t) = 2 µ δx + ~− V (x) ψ(x,t) − (1.77) ψ (x,t ) = φ¡ (x) ¢ a µ2 = ~/m Gaussian integral (2.29), (2.30) • dΓs,Q(x) exp 2πi x′,x := exp sπW x′ X − h i − Z ¡ ¢ ¡ ¡ ¢¢ π dΓs,Qx = s,Qx exp Q(x) R D − s ³ ´ Q(x)= Dx,x , W x′ = x′, Gx′ (2.28) h i h i ¡ ¢ s n ′ dΓs,Q(x) x1′ ,x ... x2′ n,x = W xi′ ,xi′ ...W xi′ ,xi′ X h i h i 2π 1 2 2n−1 2n Z ³ ´ X ¡ ¢ ³ ´ sum without repetition linear maps • Ly˜ ,x = y ,Lx (2.58) h ′ i h ′ i WY′ = WX′ L˜ , QX = QY L (Ch 3) ◦ ◦ 10 Scaling and coarse graining (section 2.5) • [u] u Slu(x)= l u l a b¡ ¢ SL[a, b[= l , l P : S £µ £ (2.83) l l/l0 · [l∞,l[∗ Jacobi operator • S′′(q) ξξ = (q) ξ, ξ (5.7) · hJ · i • b Oˆ a = O(γ)exp(iS(γ)/~) µ(γ) γ (Ch 6) h | | i D ZPab Time ordered exponential • t T exp ds A(s) (6.38) Zt0 Dynamical vector fields • A dx(t,z)= X(A) (x(t,z)) dz (t)+ Y (x(t,z)) dt (7.14) π Ψ(t,x0) := s,Q0 z exp Q0(z) φ x0 (t,z) (7.12) D 0R D · − s · ZP ³ ´ ³ X ´ A B Q0(z) := dt hABz˙ (t)˙z (t)(7.8) T Z ∂Ψ s AB ∂t = 4π h X(A) X(B) Ψ+ Y Ψ L L L (7.15) Ψ (t , x)= φ(x) 0 Homotopy • α K (b, tb; a,ta) = χ(α)K (b, tb; a,ta) (Ch 8) | | ¯ ¯ ¯ α ¯ ¯X ¯ Koszul formula ¯ ¯ • ¯ ¯ ω = Div (X) ω (11.1) LX ω · 11 Miscellaneous • det exp A = exptr A (11.48) 1 d log det A = tr A− dA (11.47) ij j −1 f := g ∂f/∂x¡ ¢ gradient (11.73) ∇g j −1 V = V divergence (11.74) ∇g | g ,j (¡V f)=¢ (divV f) gradient/divergence (11.79) |∇ − | Poisson processes • N(t) := ∞ θ (t T ) counting process (13.17) k=1 − k Density of energyP states • ρ(E)= δ (E E ) , Hψ = E ψ − n n n n n X Time ordering • φ(x )φ(x ) for j >i j i − T (φ(xj)φ(xi)) = (15.7) φ(x )φ(x ) for i > j i j − Wick (normal ordering) • operator normal ordering 2 a + a† a + a† =: : a + a† :+1 (D.1) ³ ´ ³ ´ ³ ´ functional normal ordering 1 : F (φ): := exp ∆ F (φ) (D.4) G −2 G µ ¶ functional laplacian defined by the covariance G δ δ ∆ = dvol(x) dvol(y) G(x, y) G δφ(x) δφ(y) Z Z The “Measure” (Ch 18) • µ[φ] (sdet G+[φ]) (18.3) ≈ S [φ] G+kj[φ]= δj , (18.4) i, ,k −i, G+ij[φ] = 0 when i > j (18.5) − 12 i i A I A φ = uin Aain + uin∗Aain∗ (18.18) i X i X = uout X aout + uout∗ Aaout∗ Contents Acknowledgements page 3 Symbols, Conventions and Formulary 6 Part One: The Physical and Mathematical Environment 1 The Physical and Mathematical Environment 18 1.1 The beginning 18 1.2 Integrals over function spaces 21 1.3 Operator formalism 22 1.4 A few titles 23 1.5 A tutorial in Lebesgue integration 25 1.6 Stochastic processes and promeasures 31 1.7 Fourier transformation and prodistributions 36 1.8 Planck’s blackbody radiation law 40 1.9 Imaginary time and inverse temperature 42 1.10 Feynman’s integral vs.
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