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Functional Quantization of Spinor Fields

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Functional Quantization of Spinor Fields PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 Dr. Anosh Joseph, IISER Mohali LECTURE 37 Wednesday, March 25, 2020 (Note: This is an online lecture due to COVID-19 interruption.) Topic: Functional Quantization of Spinor Fields. Functional Quantization of Spinor Fields In order to generalize the functional integral methods to spinor fields, which obey anticommutation relations, we must represent even the classical fields by anticommuting numbers. Grassmann Numbers Let us define anticommuting numbers (also called Grassmann numbers) by giving algebraic rules for manipulating them. They lead to the familiar quantum theory of the Dirac equation. A Grassmann algebra is a set of objects G that are generated by a basis fθig. These θi are Grassmann numbers, which anticommute with each other θiθj = −θjθi; (1) add commutatively θi + θj = θj + θj; (2) and can be multiplied by complex numbers, aθ 2 G for θ 2 G and a 2 C. The algebra must also have an element 0 so that θi + 0 = θi. For one θ, the most general element of the algebra is g = a + bθ; a; b 2 C; (3) since θ2 = 0. For two θ’s, the most general element is g = A + Bθ1 + Cθ2 + F θ1θ2; (4) PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 and so on. Elements of the algebra that have an even number of θi commute with all elements of the algebra, so they compose the even-graded or bosonic subalgebra. Similarly, the odd-graded or fermionic subalgebra has an odd number of θi and anticommutes within itself (but commutes with the bosonic subalgebra). The fermionic subalgebra is not closed, since θ1θ2 is bosonic. In physics our Grassmann numbers will be θ1 = (x1), θ2 = (x2), ··· , so we will have an infinite number of them. The quantities such as Lagrangian are (bosonic) elements of G. We need to find a consistent way to define integrals involving D . We want integrals to be linear Z Z Z dθ1 ··· dθn (aX + bY ) = a dθ1 ··· dθn X + b dθ1 ··· dθn Y; a; b 2 C; X; Y 2 G: (5) We do not put limits of integration on the integrals since there is only one Grassmann number R 1 in each direction. These are the analogs of the definite integrals, −∞ dxf(x), in the bosonic case. We also want integrals to be like sums so that dθ, like θ, is an anticommuting object, and so is R dθ. First consider one θ. The most general integral is Z Z Z dθ(a + bθ) = a dθ + b dθθ: (6) Since the integral is supposed to be a map from G to C, the first term vanish. We conventionally define R dθθ = 1 and so Z dθ(a + bθ) = b: (7) The obvious definition for derivatives is d (a + bθ) = b (8) dθ so integration and differentiation do the same thing on Grassmann numbers. For more θi we define Z @ @ dθ1 ··· dθn X = ··· X (9) @θ1 @θn so that Z dθ1 ··· dθn θn ··· θ1 = 1: (10) Note that we evaluate the nested integrals from the inside out Z Z dθ1dθ2 θ2θ1 = − dθ1dθ2 θ1θ2 = 1: (11) This is consistent with the order in which derivatives usually act. One important feature of these integrals is that they have the same kind of shift symmetry as the bosonic case Z 1 Z 1 dxf(x) = dxf(x + a); (12) −∞ −∞ 2 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 where a is independent of x; @xa = 0. The analog here would be Z Z dθ(A + Bθ) = dθ(A + B(θ + X)); (13) where X is any element of G that is constant with respect to θ: @θX = 0. This equality then holds the definition of integration. Since the Dirac field is complex-valued, we will work primarily with complex Grassmann num- bers, which can be built out of real and imaginary parts in the usual way. It is convenient to define complex conjugation to reverse the order of products, just like Hermitian conjugation of operators (θη)∗ ≡ η∗θ∗ = −θ∗η∗: (14) To integrate over complex Grassmann numbers, let us define θ + iθ θ − iθ θ = 1 p 2 ; θ∗ = 1 p 2 : (15) 2 2 We can now treat θ and θ∗ as independent Grassmann numbers, and adopt the convention Z dθ∗dθ(θθ∗) = 1: (16) Let us evaluate a Gaussian integral over a complex Grassmann variable Z ∗ Z Z dθ∗dθe−θ bθ = dθ∗dθ(1 − θ∗bθ) = dθ∗dθ(1 + θθ∗b) = b: (17) If θ were an ordinary complex number, this integral would equal 2π=b. The factor 2π is unim- portant; the main difference with anticommuting numbers is that the b comes out in the numerator rather than the denominator. However, if there is an additional factor of θθ∗ in the integrand, we obtain Z ∗ 1 dθ∗dθθθ∗ e−θ bθ = 1 = · b: (18) b The extra θθ∗ introduces a factor of (1=b), just as it does in an ordinary Gaussian integral. We can show that a general Gaussian integral involving a Hermitian matrix B with eigenvalues bi: Z ! Z ! Y ∗ Y P ∗ Y ∗ −θi Bij θj ∗ − i θi biθi dθi dθi e = dθi dθi e = bi = det B: (19) i i i (If θ were an ordinary number, we would have obtained (2π)n= det B.) Similarly, it is possible to show that Z ! Y ∗ ∗ ∗ −θi Bij θj −1 dθi dθi θkθl e = (det B)(B )kl: (20) i 3 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 In general, except for the determinant being in the numerator rather than the denominator, Gaussian integrals over Grassmann variables behave exactly like Gaussian integrals over ordinary variables. The Dirac Propagator A Grassmann field is a function of spacetime whose values are anticommuting numbers. We can define a Grassmann field (x) in terms of any set of orthonormal basis functions X (x) = iφi(x): (21) i The basis functions are ordinary c-number functions, while the coefficients i are Grassmann numbers. To describe the Dirac field, we take the φi to be a basis of four-component spinors. We now have all the machinery needed to evaluate functional integrals, and hence correlation functions, involving fermions. For example, the Dirac two-point function is given by R R 4 D D exp i d x (i@= − m) (x1) (x2) h0jT (x1) (x2)j0i = : (22) R D D exp i R d4x (i@= − m) (We write D instead of D ∗, for convenience; the two are unitarily equivalent.) Evaluating this in Fourier space, we find that Z d4k ie−ik·(x1−x2) h0jT (x ) (x )j0i = S (x − x ) = : (23) 1 2 F 1 2 (2π)4 k= − m + i Higher correlation functions of free Dirac fields can be evaluated in a similar manner. Generating Functional for the Dirac Field As with the Klein-Gordon field, we can alternatively derive the Feynman rules for the free Dirac theory by means of a generating functional. We define the Dirac generating functional as Z Z Z[η; η] ≡ D D exp i d4x (i@= − m) + η + η ; (24) where η(x) is a Grassmann valued source field. We can shift (x) to complete the square, to derive the simpler expression Z 4 4 Z[η; η] = Z0 · exp − d xd yη(x)SF (x − y)η(y) ; (25) where, as before, Z0 is the value of the generating functional with the external sources set to zero. 4 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 To obtain correlation functions, we will differentiate Z with respect to η and η. The sign conventions we adopted for derivatives with respect to Grassmann numbers are d d θη = − ηθ = −θ; (26) dη dη for anticommuting numbers η and θ. Then, the two-point function is given by δ δ −1 h0jT (x1) (x2)j0i = SF (x1 − x2) = Z0 −i +i Z[η; η] : (27) δη(x1) δη(x2) η,η=0 Higher correlation functions can also be evaluated in a similar way. References [1] M. E. Peskin and D. Schroeder, Introduction to Quantum Field Theory, Westview Press (1995). [2] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013). 5 / 5.
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