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November 30, 2016 Theoretical Exercises 2016 Thomas Becher Albert Einstein Center for Fundamental Physics Institute for Theoretical Physics University of Bern Contents 1 Gaussian integrals and Wick's theorem 1 1.1 Gaussian integrals . .1 1.2 Generating function, Wick's theorem . .2 2 Integrals over Grassmann variables 4 3 Dirac Algebra 6 3.1 Levi-Civita symbol . .6 3.2 Dirac Algebra . .8 4 Klein-Gordon, Maxwell and Proca equations 11 4.1 Klein Gordon equation . 11 4.2 Electromagnetic field in relativistic notation . 11 4.3 Proca equation . 12 4.4 Gauge invariance . 13 5 Spinors and Dirac equation 15 5.1 Lorentz transformation of spinor fields . 15 5.2 Dirac Lagrangian and Dirac equation . 19 5.3 Plane-wave solutions . 19 6 Non-relativistic limit of the Dirac equation 22 6.1 Dirac equation in the presence of an electromagnetic field . 22 6.2 Magnetic moment of the electron . 23 7 Interacting fields in perturbation theory 25 7.1 Relativistically invariant interaction Lagrangians . 25 7.2 Perturbation theory . 27 7.3 Wick's theorem . 28 7.4 Feynman rules . 30 7.4.1 φ4-theory . 30 7.4.2 QED . 31 7.4.3 QED with an extra neutral scalar S .................... 32 8 Computation of scattering cross sections 33 8.1 Reduction formula and S-matrix . 33 8.2 Cross section and decay rate . 35 9 Diphoton production and a scalar resonance 37 9.1 Decays of the scalar S ............................... 38 9.2 Diphoton production . 39 9.3 Finite-width effects . 41 9.4 Data analysis . 42 2 1 Gaussian integrals and Wick's theorem 1.1 Gaussian integrals Exercise 1.1. Compute the basic, one-dimensional Gaussian integral and show that it is given by 2 π 1 dx e λx = ; λ > 0: (1) − λ Z−∞ r The standard trick to obtain this result is to take the square and use polar coordinates to evaluate the integral. Notice that λ can be complex, as long as Re(λ) > 0. Exercise 1.2. Show that 2 2 π a 1 dx e λx +ax = exp ; λ > 0: (2) − λ 4λ Z−∞ r Next, we turn to n-dimensional Gaussian integrals. We consider an element x of Rn and a symmetric n n matrix M and define a quadratic form xT Mx (we view x as a column matrix × to avoid writing vector arrows). Integrating over Rn, we obtain the Gaussian integral n T 1 d x exp x Mx = − (3) − N Z Exercise 1.3. Show that det M = : (4) n N r π What are the conditions on the matrix M for the integral (3) to exist? Exercise 1.4. Similarly, show that for a Rn 2 n T T 1 T 1 d x exp x Mx + a x = exp a M − a : (5) N − 4 Z Gaussian integrals appear in many areas of physics, in particular in statistical physics and in path-integral quantization. In Feynman's path-integral formulation of quantum mechanics, one needs to compute an integral of the form x ; t x ; t = x(t) eiS[x(t)] (6) h f f j 0 i D Z to obtain the probability amplitude that a particle which was at point x0 at time t0 can be found at point x1 at time t1, where S[x(t)] is the action associated with the path x(t) and the symbol x(t) indicates that one should sum over all paths x(t) starting at x(t0) = x0 and ending atDx(t ) = x . R f f 1 As it stands, it is not clear what (6) means. To define expression (6), one discretizes time t = i ∆ such than t = t . After which the integral over all paths takes the form i · n+1 f n x(t) dx = dnx (7) D −! i i=1 Z Y Z Z where xi = x(ti) is the position after i time steps. We see that we encounter integrals over Rn as in (3). To evaluate the oscillatory expression (6), one first computes it for imaginary (Euclidean) time t = iτ with τ R. Performing this so-called Wick rotation leads to − 2 iS[x(t)] SE[x(τ)] so that one can work with a real exponent as in (3). After performing the integrals! − and taking after taking ∆ 0 one then analytically continues result back to physical time-values . Doing so, the path! integral for a harmonic oscillator (or a free particle) boils down to the evaluation of Gaussian integrals as in (3). All these steps will be explained and carried out in the Advanced Concepts course, in these exercises we focus on the computation of the Gaussian integrals. The path integral formulation plays an important role in QFT, where it is also the basis for numerical computations. In field theories one integrates over all field configurations φ(t; ~x) instead of the path x(t). In discretized form, one then integrates over the field values at points φi φ(ti; ~xi). The discrete set of points (ti; ~xi) is called a lattice and the to obtain the continuum≡ result one takes the limit where the distance between lattice points (the lattice spacing) goes to zero. 1.2 Generating function, Wick's theorem If we include the normalization factor , we can view the integrand in (3) N ρ(x) = exp xT Mx ; (8) N − as a probability distribution in Rn since it is normalized and strictly positive as long as M is a real, symmetric and positive1 matrix. We can then compute expectation values A(x) = dnx ρ(x) A(X) : (9) h i Z The m-point correlation functions x : : : x ; (10) h i1 im i play an important role when one computes path integrals and we will now analyze them in detail. The result is the Wick theorem, which provides the basis for perturbative computations in QFT. To obtain the expectation values (10), let us consider 1 (b) = ebixi = dnx ρ(x)ebixi = b : : : b dnx ρ(x) x : : : x : (11) Z h i m! i1 im i1 im m 0 Z X≥ Z 1All its eigenvalues are strictly positive. 2 The quantity (b) is the generating function of the moments of the probability distribution ρ(x): Z 1 (b) = b : : : b x : : : x : (12) Z n! i1 im h i1 im i m 0 X≥ The inverse relation can be written @ @ x : : : x = ::: (b) : (13) h i1 im i @b @b Z i1 im b=0 In general, for an arbitrary probability density ρ(x), (b) cannot be exactly calculated. But it can easily be evaluated for the Gaussian probabilityZ density. According to result (5), the generating function of the moments of the Gaussian distribution is 1 (b) = dnx exp xT Mx + bT x Z N −2 Z = exp(bT x) (14) h i 1 = exp bT M 1b : 2 − The inverse relation is @ @ 1 1 x : : : x = ::: exp b (M − ) b : (15) h i1 im i @b @b 2 i ij j i1 im b=0 Exercise 1.5. Prove that: @ 1 exp b (M 1) b = (M 1) b exp 1 b (M 1) b ; @b 2 i − ij j − i1j j 2 i − ij j i1 @ @ 1 1 1 1 1 exp bi(M − )ijbj = (M − )i1i2 + (M − )i1jbj(M − )i2kbk @bi1 @bi2 2 h i 1 1 exp b (M − ) b : · 2 j jk k @ @ 1 1 Exercise 1.6. Find a general expression for ::: exp bi(M − )ijbj . Looking at the @bi1 @bin 2 explicit expressions for the lowest few derivatives, you should observe a simple pattern, which can then be established using induction. Exercise 1.7. Use the previous results at b = 0, as in the inverse relation (15), to verify that 1 x x = (M − ) ; h i1 i2 i i1i2 1 1 1 1 x x x x = (M − ) (M − ) + (M − ) (M − ) (16) h i1 i2 i3 i4 i i1i2 i3i4 i1i3 i2i4 1 1 +(M − )i1i4 (M − )i2i3 ; 3 and the correlators of an odd number of points vanish, x : : : x = 0 : h i1 i2m+1 i Exercise 1.8. Derive Wick's theorem: xi1 : : : xi2m = xk1 xk2 ::: xk2m 1 xk2m ; (17) h i h i h − i P X where the sum is over all pairings P , i.e. all possible ways to group the indices the indices i1; i2; : : : ; i2m into m pairs (k1; k2);:::; (k2m 1; k2m). The theorem states for the Gaussian integral all higher-point correlators reduce− to products of the 2-point correlation function, which is given by the inverse of the matrix in the exponent of the Gaussian integral 1 x x = (M − ) (18) h i1 i2 i i1i2 as we have shown above. (As usual in mathematics, the proof of the final theorem is trivial after you have prepared it with a highly nontrivial proof of a lemma, in our case the general form of the derivative in Exercise 1.6.) 2 Integrals over Grassmann variables To construct a consistent quantum theory of fermionic fields, the field operators must fulfil anti-commutation relations (3) (x); y (y) = δ (~x ~y)δ ; (19) α β − αβ α(x); β(y) = αy (x); βy (y) = 0 ; at equal times x0 = y0. To construct a path-integral representation of a such a theory, one uses anti-commuting variables, which are also called Grassmann numbers. They are introduced and explored in this part of the exercises. We consider a set of n such numbers gi, i = 1 : : : n, which fulfil the Grassmann algebra ηi; ηj = 0 ; (20) 2 implying that ηi = 0, which makes the algebra extremely simple.
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