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Compendium of Useful Formulas

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Compendium of Useful Formulas Preprint typeset in JHEP style - HYPER VERSION Compendium of useful formulas Matthew Headrick Abstract: Almost everything I know. Contents 1. Miscellaneous math 3 1.1 A matrix identity 3 1.2 Campbell-Baker-Hausdorff 3 1.3 Pauli matrices 3 1.4 Level repulsion 3 1.5 The Legendre transform 3 1.6 Gaussian integrals 5 1.7 Saddle-point approximation 5 1.8 Cauchy principal value 6 1.9 A few Fourier transforms 7 2. Differential geometry 7 2.1 Diffeomorphisms vs. coordinate tranformations 7 2.2 Connection and curvatures 8 2.3 Geodesic deviation equation 9 2.4 Forms 10 2.5 Harmonic forms 11 2.6 Vielbein formalism 12 2.7 Lie derivatives and Killing fields 13 2.8 Weyl transformations 14 2.9 Metric perturbations 14 2.10 Warped products 14 2.11 Flat fibrations 15 2.12 Gauss-Bonnet theorems 16 2.13 Hypersurfaces 17 2.14 General submanifolds 21 2.15 K¨ahlergeometry 21 2.16 Poincar´eduality 23 2.17 Coordinates for S2 24 2.18 The Hopf fibration 25 2.19 Unit sphere 27 3. Classical mechanics 27 3.1 Lagrangian and Hamiltonian formulations 27 3.2 Symmetries 29 3.3 Noether's theorem for higher-order actions 31 3.4 Gauge symmetries 32 3.5 Hamilton-Jacobi theory 34 { 1 { 4. Gauge theories 36 4.1 Generalities 36 4.2 Standard Model 37 5. General relativity 40 5.1 Dimensions 40 5.2 Fundamentals 40 5.3 Spherically symmetric solutions 41 5.4 Cauchy problem 42 5.5 Gravitational waves 44 5.6 Israel junction condition 46 5.7 Boundary term 47 5.8 Dimensional reductions 49 6. String theory & CFT 50 6.1 Complex coordinates 50 6.2 CFT stress tensor 50 6.3 Liouville action 51 6.4 Sigma model 51 6.5 Spacetime action 52 6.6 Dimensional reduction and Buscher duality 53 6.7 More dualities 55 6.8 Orbifolds and quotient groups 56 7. Quantum information theory 56 7.1 Von Neumann entropy 56 7.2 Relative entropy 57 7.3 R´enyi entropy 58 7.4 Mixing-enhancing 59 7.5 Joint systems 60 8. Superselection rules 62 8.1 Joint systems 62 8.2 Joint systems and fermionic operators 63 9. Dimensions of physical quantities 64 { 2 { 1. Miscellaneous math 1.1 A matrix identity If fgabg is a symmetric matrix, then " # x vb 1 det = x det gab − wavb = α detfgabg; (1.1) wa gab x −1 " # " c # x vb 1 1 −v = b bc b c ; (1.2) wa gab α −w αg + w v 1 −1 1 g − w v = gbc + wbvc (1.3) ab x a b α bc c bc b bc bc where g is the inverse of gab, v = vbg , w = g wc, and α = x − vbg wc. 1.2 Campbell-Baker-Hausdorff If [A; B] commutes with A and B, then [A; eB] = [A; B]eB (1.4) eABe−A = B + [A; B] (1.5) eAeB = e[A;B]eBeA (1.6) eA+B = e−[A;B]=2eAeB: (1.7) 1.3 Pauli matrices " # " # " # 0 1 0 −i 1 0 σ = ; σ = ; σ = : (1.8) 1 1 0 2 i 0 3 0 −1 1.4 Level repulsion In the space of n × n Hermitian matrices, the matrices with two degenerate eigenvalues lie on a (real) co-dimension 3 submanifold. Therefore, a 1-parameter family of matrices H(λ) will not generically intersect the submanifold: as λ is varied, the eigenvalues will move around, but will not cross each other. However, if there is a symmetry, i.e. another Hermitian matrix S that commutes with H(λ) for all λ, then the eigenvalues of H(λ) belonging to different eigenspaces of S are independent, and will generically cross. 1.5 The Legendre transform Let f(x) be a function on a vector space that is bounded below and goes to plus infinity faster than linearly in every direction. The Laplace transform of e−f(x) is eg(k), where g(k) is a function on the dual vector space: Z i eg(k) = dx ekix −f(x): (1.9) { 3 { It is easy to show that g(k) is convex. The Legendre transform of f(x) is the saddle-point approximation to g(k). Assume first that f is strictly convex and continuously differentiable. Then for every k the function i f(x) − kix has a unique (local and global) minimum, where @f = k : (1.10) @xi i In other words, we have a bijection between x and k. The saddle point approximation to the integral (1.9) is the following function of k. i g(k) = kix − f(x) @f : (1.11) =ki @xi This will also be strictly convex and continuously differentiable. The inverse map is given by i x = @g=@ki, and we have i i f(x) = kix − g(k) @f = kix − g(k) @g i : (1.12) =ki =x @xi @ki In other words, the Legendre transform is its own inverse. i If f is not convex, then for some values of k, f(x) − kix will have multiple local minima (and saddle points). However, it is the global minimum that dominates the integral (1.9). The x value of this global minimum will jump discontinuously as a function of k. This shows that the Legendre transform g(k) will have a discontinuous first derivative. In fact, the global minimum won't change at all if we replace f(x) with its convex hull (as in the Maxwell construction). Therefore, two functions with the same convex hull will have the same Legendre transform. If f(x) has a discontinuous first derivative (at a convex point), then the global minimum i of f(x) − kix will be at the same value of x over a range of values of k. Over that range g(k) will be a linear function, and therefore won't be strictly convex. In summary, on convex functions, having a range over which the function is linear (i.e. not being strictly convex) makes its Legendre transform have a discontinuous first derivative, and vice versa; the Legendre transform continues to be its own inverse. The double Legendre transform of a non-convex function returns its convex hull. If f depends on an auxiliary parameter y, then g will also depend on y. Their derivatives with respect to y obey a simple relation: @g @f = − : (1.13) @y @y { 4 { 1.6 Gaussian integrals Z 2 dx e−ax =2 = (2π)1=2a−1=2 (1.14) Z 2 dx x2e−ax =2 = (2π)1=2a−3=2 (1.15) Z 2 dx x4e−ax =2 = 3(2π)1=2a−5=2 (1.16) Z 2 dx x6e−ax =2 = 15(2π)1=2a−7=2 (1.17) n+1=2 Z 2 1 2 dx x2ne−ax =2 = (2n − 1)!! (2π)1=2a−(n+1=2) = Γ n + (1.18) 2 a Z i j dmx e−Aij x x = (2π)m=2(det A)−1=2 ; (where A is symmetric) (1.19) 1.7 Saddle-point approximation Here we treat only the simplest case of a single real integral of a real function. Let f; g be real analytic functions, with f be bounded below with a single global minimum at x0. Set 00 0 f0 = f(x0) ; f2 = f (x0) ; g0 = g(x0) ; g1 = g (x0) ; etc. (1.20) 0 (of course, f (x0) = 0). We will also assume f2; g0 6= 0. Taking to be a small (positive) parameter, we derive the following approximation, working to the necessary order to get to order in the final answer: Z dx g(x)e−f(x)/ Z = dx g(x) 1 1 1 1 × exp − f + f (x − x )2 + f (x − x )3 + f (x − x )4 + O(x − x )5 0 2 2 0 6 3 0 24 4 0 0 Z 1 = e−f0/1=2 dy g + 1=2g y + g y2 + O(3=2) 0 1 2 2 1 1 1 × exp − f y2 − 1=2f y3 − f y4 + O(3=2) 2 2 6 3 24 4 Z g 1 g −f0/ 1=2 1=2 1 2 2 3=2 = e g0 dy 1 + y + y + O( ) g0 2 g0 1 1 1 2 × 1 − 1=2f y3 − f y4 + f 2y6 + O(3=2) e−f2y =2 6 3 24 4 72 3 2 Z g f g f f 2 −f0/ 1=2 2 2 3 1 4 4 4 3 6 2 −f2y =2 = e g0 dy 1 + y − y − y + y + O( ) e 2g0 6g0 24 72 2π 1=2 g f g f 5f 2 −f0/ 1=2 2 3 1 4 3 2 = e g0 1 + − 2 − 2 + 3 + O( ) ; (1.21) f2 2g0f2 2g0f2 8f2 24f2 { 5 { −1=2 In the second equality we changed the integration variable to y = (x−x0). In the fourth equality we dropped odd powers of y, since those will integrate to 0. The result is conveniently written as a decreasing series in for its logarithm: Z 2 2 −f(x)/ f0 1 2πg0 g2 f3g1 f4 5f3 2 ln dx g(x)e = − + ln + − 2 − 2 + 3 + O( ) : 2 f2 2g0f2 2g0f2 8f2 24f2 (1.22) In the case g0 = 0 (but g2 − f3g1=f2 6= 0), one gets instead Z π 1=2 f g −f(x)/ −f0/ 3=2 3 1 dx g(x)e = e 3 g2 − + O() (1.23) 2f2 f2 Z 3 −f(x)/ f0 1 π f3g1 ln dx g(x)e = − + ln 3 + ln g2 − + O() : (1.24) 2 2f2 f2 If there is another local minimum, or if the integration region is bounded (but includes x0 in its interior), then this series is asymptotic, and there are non-perturbative corrections −f(x )/ (f −f(x ))/ to (1.21) that go like e 1 (or to (1.22) like e 0 1 ), where x1 is the local minimum (other than x0) or integration endpoint with the lowest value of f.
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