Some Facts About Generalized Functions

Appendix A Some Facts About Generalized Functions

We recall here some basic facts and formulas from the theory of generalized functions. There are many mathematical textbooks on this subject. Physicists may find useful concise texts, for example Chaps. 2÷4 in [12], or Chaps. 2÷3 in [11]. Com- prehensive introduction to the subject can be found in [2]. The generalized functions that appear in field theory are of the so called Schwartz class, denoted as S(Rn) or S∗(Rn). The reason is that all such generalized functions have a Fourier transform. The generalized function of the Schwartz class1 (g. f.) is, by definition, a linear and continuous functional on the Schwartz space of functions, denoted as S(Rn). Elements of S(Rn) are called test functions. They are complex ∞ valued functions on Rn of the C class. Moreover, it is assumed that such functions, and all their derivatives, vanish in the limit |x|= (x1)2 + (x2)2 + ...+ (xn)2 → ∞, also when multiplied by any finite order polynomial in the variables x1,...,xn. Here x denotes arbitrary point in Rn and x1,...,xn are its Cartesian coordinates. The space S(Rn) is endowed with a topology, but we shall not describe it here. 2 Examples of test functions from the space S(R1) include e−ax and 1/ cosh(ax), where a > 0 is a real constant. On the other hand, (1 + x2)−1 is not test function from S(R1). The value of a generalized function F ∈ S∗(Rn) on a test function f ∈ S(Rn) is denoted in mathematical literature as F(x), f (x), but in physics the most popular is the misleading notation dn xF(x) f (x), e.g., dn x δ(x) f (x) in the case of the Dirac delta. One should keep in mind that the integral here is merely a symbol that replaces , from the mathematical notation—it is not the true integral. It may hap- pen however, that a generalized function is represented by an ordinary function F(x) such that the true integral dn xF(x) f (x) exists for all f ∈ S(Rn). Such F is called regular g.f. For example, the step function(x), x ∈ R1, is a regular generalized ∗( 1) ∞ ( ) ( ) = ∞ ( ) function from S R , because the integral −∞dx x f x 0 dx f x exists for every f ∈ S(R1). We show the integration range when we deal with the true integral. The Dirac delta δ(x) is the prominent example of non regular g.f. In the mathematical notation its definition has the form δ(x), f (x)= f (0).

1 Other names are also used: distribution for generalized function, and tempered distribution for generalized functions of the Schwartz class.

H. Arod´z, L. Hadasz, Lectures on Classical and Quantum Theory of Fields, 343 DOI 10.1007/978-3-642-15624-3, C Springer-Verlag Berlin Heidelberg 2010 344 A Some Facts About Generalized Functions

The derivative ∂i F of a g.f. F is deﬁned as follows: n n d x ∂i F(x) f (x) =− d xF(x)∂i f (x) for all test functions f . One should remember that this is the deﬁnition, and not the formula of integration by parts. For example, d(x) ∞ df(x) ∞ df(x) dx f (x) =− dx (x) =− dx = f (0), dx −∞ dx 0 dx

hence (x) = δ(x). Derivative of g.f. always exists and is a generalized function. ˜( ) = ( π)n/2 n ( ) ( ) The Fourier transform f k 2 Rn d x exp ikx f x of a test function f also is a test function from the space S(Rn). The g.f. F˜(x) ∈ S∗(Rn) such that for every f ∈ S(Rn) dn x F˜(x) f (x) = dnkF(k) f˜(k), is called the Fourier transform of the g.f. F. It exists for any F ∈ S∗(Rn). The oper- ation of taking the Fourier transform is continuous with respect to F. This property is used in order to facilitate computation of the Fourier transform of (x)—we ﬁrst compute the Fourier transform of e−εx (x), where ε>0, and take the limit ε → 0+ at the end.2 Thus,

∞ −ε dx (˜ x) f (x) = lim dke k(k) f˜(k) ε→0+ −∞ ∞ ∞ ∞ 1 −ε i 1 = lim √ dk dx eikx k f (x) = lim √ dx f (x). ε→0+ 2π 0 −∞ ε→0+ 2π −∞ x + iε

The r.h.s. of this formula deﬁnes the g.f. denoted as √i 1 . Therefore, 2π x+i0+

i 1 (˜ x) = √ . 2π x + i0+

1 ε> Note that the g.f. x+iε is regular if 0. One can prove that

1 1 = P − iπδ(x), x + i0+ x

1 where the principal value distribution P x is deﬁned as

2 The notation ε → 0+ means that ε = 0 is approached from the side ε>0. A Some Facts About Generalized Functions 345 1 −ε f (x) ∞ f (x) dx P f (x) = lim dx + dx x ε→0+ −∞ x ε x

(it is not regular). The result for (˜ x) obtained above is often written in the form ∞ 1 dp eipx = iP + πδ(x). 0 x The form of a generalized function can be probed only with test functions. Because there is no test function with support consisting of just a single-point, it is not possible to tell what is the value of g.f. at a given point. One can however ( ) ∈ n check whether a g.f., say F x , is constant in a vicinity Vx0 of a point x0 R —it is sufﬁcient to show that all the ﬁrst derivatives of F(x) vanish in that vicinity, i.e., that

∂i F(x) f (x) = 0

⊂ n δ( ) ∈ for every test function that has its support in Vx0 R . For example, for x S∗(R1) one may say that δ(x) = 0oneveryinterval(a, b) that does not contain 0, and that δ(x) = 0atx = 0, but not that δ(1) = 0. A consequence of the lack of definite value at a single point is that there is no general definition of product of generalized functions. We know the gener- ∗ n ∗ n alized functions F1(x) ∈ S (R ), F2(x) ∈ S (R ) if we know the values of n n n d xF1(x) f (x), d xF2(x) f (x) for every f ∈ S(R ). It is not possible to n infer from this what values should have d xF1(x)F2(x) f (x).Onlyinsome special cases, e.g., for certain regular generalized functions, such product can be defined. In particular, there is no problem with multiplication by an ordinary function ψ(x), provided that ψ(x) f (x) ∈ S(Rn) for every f ∈ S(Rn). Then, the product ψ(x)F(x) is the g.f. defined by the formula dn x (ψ(x)F(x)) f (x) = dn xF(x)(ψ(x) f (x)).

For example, if k is a fixed real number, eikxδ(x) is a generalized function, while xaδ(x) with non integer constant a > 0 is not (not all functions xa f (x) belong to S(R1) because of the problem with derivatives at x = 0). On the other hand, there is no difficulty with a product of generalized functions ∗ ∗ with different arguments. If F(x) ∈ S (Rn) and G(y) ∈ S (Rm), then we know dn xF(x) f (x) and dm yG(y)g(y) for all f ∈ S(Rn), g ∈ S(Rm). The generalized function H(x, y) = F(x)G(y) ∈ S∗(Rn+m) is defined by its action on the test functions h(x, y) ∈ S(Rn+m) of the form h(x, y) = f (x)g(y), namely dn xdm yH(x, y)h(x, y) = dn xF(x) f (x) dm yG(y)g(y). 346 A Some Facts About Generalized Functions

Such factorized test functions f (x)g(y) form a subset of S(Rn+m) that is sufﬁciently large to uniquely determine H(x, y) on the whole space S(Rn+m).An example: if x, y ∈ R1 are independent variables, then δ(x)δ(y) ∈ S∗(R2). Finally, let us consider the question whether ∞ 0 dx δ(x) =? dx δ(x) + dx δ(x), 0 −∞ or, in a more meaningful form, whether

? δ(x) = 1δ(x) = ((x) + (−x))δ(x) = (x)δ(x) + (−x)δ(x).

The answer is that such a formula is wrong, because the products (x)δ(x), (−x)δ(x) are not deﬁned. The way to correct the splitting consists in replacing (±x) with two smooth functions θ1(x), θ2(x) that obey the condition θ1(x) + θ2(x) = 1, and resemble (x), (−x), respectively. Moreover, these functions 1 1 should be such that θi (x) f (x) ∈ S(R ), i = 1, 2, for every f ∈ S(R ). Then we may safely write

F(x) = θ1(x)F(x) + θ2(x)F(x) for any F(x) ∈ S∗(R1). Bibliography

1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Applied Mathematics Series 55. National Bureau of Standards, Washington, DC (1964) 2. Vladimirov, V.S.: Methods of the Theory of Generalized Functions. CRC Press. Boca Raton, FL (2002) 3. Bogoljubov, N.N., Shirkov, D.V.: Introduction to the Theory of Quantized Fields. Wiley- Interscience. New York, NY, London, Sydney, Toronto, ON (1980). 4. Jackiw, R., Manton, N.: Symmetries and conservation laws in gauge theories. Ann. Phys. 127, 257 (1980). 5. Schweber, S.S.: An Introduction to Relativistic Quantum Field Theory. Row, Peterson and Co., Evanston, IL, Elmsford, NY (1961) 6. Henneaux, M., Teitelboim, C.: Quantization of Gauge Systems. Princeton University Press, Princeton, NJ (1992) 7. Faddeev, L.D., Jackiw, R.: Hamiltonian reduction of unconstrained and constrained systems. Phys.Rev.Lett.60, 1692 (1988) 8. Sakita, B.: Quantum Theory of Many-Variable Systems and Fields. World Scientiﬁc, Singa- pore (1985) 9. Sterman, G.: An Introduction to Quantum Field Theory. Cambridge University Press, Cam- bridge, New York, NY, Melbourne (1993) 10. Weinberg, S.: The Quantum Theory of Fields, vol. I. Cambridge University Press, Cambridge, New York, NY, Melbourne (1995) 11. Bogoljubov, N.N., Logunov, A.A., Oksak, A.I., Todorov, I.T. General Principles of Quantum Field Theory. Kluwer, Dordrecht (1990) 12. Richtmyer, R.D.: Principles of Advanced Mathematical Physics, vol. 1. Springer. New York, NY, Heidelberg, Berlin (1978)

347

Index

A E Abelian Higgs model, 81 Electromagnetic field, 7 Action functional, 22 Energy-momentum tensor, 40 Annihilation operator, 127 Equal time canonical commutation relations, Anomaly equation, 331 117 Antiparticle, 141 Equal-time anticommutation relations, 132 Axial anomaly equation, 340 Euler–Lagrange equations, 22 External vertex, 170 B Bare coupling constant, 204 F ( ) Bare mass, 204 F p , 169 Bianchi identity, 78 Faddeev–Popov–DeWitt determinant, 279 BPHZ subtractions, 196 Feynman diagrams, 170 BRST transformation, 290 Fierz identity, 320 Finite mass counterterm, 253 Fock space, 122 C Free propagator F , 163 Callan–Symanzik equation, 217, 224 Functional derivative, 22 Cancelation of anomaly, 333 Fundamental fields, 7 Chiral rotations, 335 Chiral superfield, 309 G Classical Dirac field, 95 Gauge condition Cocycle, 231 Coulomb, 9 Complex scalar field, 49 general, 278 Connected graphs, 183 Lorentz, 281 Continuity equation, 30 Gauge fixing term, 285 Counterterm, 194 Gauge group, 65 Covariant derivative, 66 Gauge invariant point splitting, 327 Creation operator, 127 Gauge potentials, 8 Gell-Mann–Low D formula, 160 Dimensional transmutation, 223 β function, 217 Dirac Generating functional Z[ j], 161 bispinor, 87 Ghost fields, 281 equation, 87 Goldstone relations, 87 field, 54 sea, 138 model, 51 vacuum, 138 Grassmann derivative, 104 Domain walls, 45 Grassmann integration, 271

349 350 Index

Green’s functions, 157 O Gribov problem, 282 One-particle irreducible (1PI) graphs, 184

H P Higgs mechanism, 82 Parallel transport, 67 Hilbert space realization, 120 Particle interpretation, 127 Particle number operator, 128 Path integral I for generating functional Z[ j], 266 Interaction picture, 154 for evolution operator, 265 Internal vertex, 170 Path-ordered exponential, 73 Invariant volume element for SU(N), 278 Pauli–Jordan functions, 16 Pauli–Villars regularization, 189 J Pendulums, 1 Jacobi identity, 74 Perturbatively renormalizable model, 187 Photons, 147 K 1PI graphs, 184 Killing four-vectors, 25 1PI irreducible Green’s functions, 205 Klein–Gordon equation, 12 Poincaré group, 35 Primitively divergent graphs, 196 L Proca equation, 82 L eft-mover field, 322 Proca field, 83 Lagrangian, 22 Projective representation of the Poincaré Lie algebra of SU(N), 72 group, 231 Lie derivative, 25 Proper vertices, 205 Local U(1) group, 65 Local quantum field, 117 R γ Loop graphs, 182 5 matrix, 96 Lorentz Ray in Hilbert space, 228 group, 35 Renormalization condition, 193 transformations, 34 Renormalization group trajectory, 213 Retarded Green’s function, 14 Right-mover field, 322 M Running coupling constant, 212 Majorana condition, 103 Majorana field, 102 S Majorana representation of Dirac matrices, 90 Scalar electrodynamics, 70 Maxwell equations, 7 Sinus-Gordon MI renormalization scheme, 221 antisoliton, 6 Minimal coupling prescription, 67 equation, 3 Multiparticle spectral function, 245 soliton, 6 Multiplicative renormalization, 204 SL(2, C) group, 92 Slavnov–Taylor identity, 292 N Spectral decomposition for G(2), 245 Nambu–Goto string, 30 Spin(4) group, 92 Natural units, 63 Spontaneous symmetry breaking, 43 Noether’s Stationary action principle, 22 identity, 29 SU(N) group, 72 theorem, 24 Subtraction point, 193 Non-Abelian field strength tensor, 75 Superficial degree of divergence, 186 Non-Abelian gauge field, 72 Superfield, 305 Nonrenormalizable model, 187 Superrenormalizable model, 187 Normal ordered interaction, 173 Superspace, 305 Normal ordering, 122 Supersymmetry algebra, 295 Index 351

Symmetric subtraction point, 194 V Symmetry of quantum systems, 228 Vacuum bubbles, 175 Symmetry transformation Vacuum state, 122 classical, 25 Vector ﬁeld, 63 quantum, 235 Vortex, 57

W Wess–Zumino model, 312 T Weyl ﬁelds, 97 Topological charge, 59 Weyl spinors, 100 Tree graphs, 182 Wick formula, 162 Wick rotation, 192 winding number, 56 U U(1) group, 50 Y U(N) group, 71 Yang–Mills equation, 76