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Abelian Gauge Theories

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Quantum Field Theory I

Part of the book series: Graduate Texts in Physics ((GTP))

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Abstract

The major theme of quantum field theory is the development of a unified theory that may be used to describe nature from microscopic to cosmological distances. Quantum field theory was born 90 years ago, when quantum theory met relativity, and has captured the hearts of the brightest theoretical physicists in the world. It is still going strong. It has gone through various stages, met various obstacles on the way, and has been struggling to provide us with a coherent description of nature in spite of the “patchwork” of seemingly different approaches that have appeared during the last 40 years or so, but still all, with the common goal of unification.

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Notes

  1. 1.

    A geometrical description is set up for the development of abelian and non-abelian gauge theories in a unified manner in Sect. 6.1 and may be beneficial to the reader.

  2. 2.

    It is worth knowing that the name “photon” was coined by Lewis [43].

  3. 3.

    See also Eq. (2.2.11).

  4. 4.

    See (4.7.120), (4.7.121), (2.2.17), (2.2.18), (2.2.19), (2.2.20), (2.2.21) and (2.2.22).

  5. 5.

    A reader who is not familiar with this expression may find a derivation of it in Manoukian [56], p. 84. See also (2.2.11).

  6. 6.

    We follow Schwinger’s elegant construction [70].

  7. 7.

    Since the gauge constraint is now a derived one, one may vary all the components of the vector field independently.

  8. 8.

    For J μ = 0.

  9. 9.

    Schwinger [72] and Manoukian [50].

  10. 10.

    D +  < (x, x  ​′) stands for D +(x, x  ​′) for x 0 < x  ​′ 0.

  11. 11.

    See, e.g. Kenneth et al. [36] and Milton et al. [64].

  12. 12.

    See, e.g., Elizalde et al. [20] and Elizalde [19].

  13. 13.

    See Vol. II: Quantum Field Theory II: Introductions to Quantum Gravity, Supersymmetry, and String Theory, (2016), Springer.

  14. 14.

    This was conveniently introduced by Schwinger [70].

  15. 15.

    See, e.g., Manoukian [57].

  16. 16.

    See, e.g., op. cit.

  17. 17.

    See discussion above Eq. (3.3.39).

  18. 18.

    Note that the scaling factors are not ɛ 2, 1∕μ 2, respectively, as one may naïvely expect. The reason is that functional differentiation of the action, with respect to the vector potential, involving the quadratic terms ɛ F 0 i F 0 i, F ij F ij∕μ, generate the linear terms corresponding to the electric and magnetic fields components which are just needed in deriving Maxwell’s equations.

  19. 19.

    For these additional details, see, e.g., Manoukian and Charuchittapan [59].

  20. 20.

    See, e.g., Fazio et al. [24], Pinsky et al. [66], and McNulty et al. [62].

  21. 21.

    Recall from Sect. 3.9, that within a time ordered product the Dirac fields anti-commute.

  22. 22.

    The path integral version of (5.7.20) is derived in the next subsection.

  23. 23.

    The expressions in (5.7.20) and (5.7.27) are appropriately referred to as generating functionals. As we have no occasion to deal with subtleties in defining larger vector spaces to accommodate covariant gauges, we will not go into such technicalities here.

  24. 24.

    Note that the first order contribution involves the expression: \(\text{Tr}\,[\,\gamma \,^{\mu }\,S_{+}(0)\,]\, = -\text{Tr}\,[\,\gamma \,^{\mu }\,\gamma \,^{\sigma }]\int \big((\mathrm{d}p)/(2\pi )^{4}\big)p_{\sigma }/(\,p^{2} + m_{0}^{2}) = 0\), as it has an odd integrand in p.

  25. 25.

    These processes are assumed to involve no external sources or external potentials.

    Fig. 5.3
    figure 3

    (a ) Consider the diagram describing the scattering process of an electron, experiencing a change of its three-momentum, via the exchange of a (virtual) photon with the remaining part of the diagram (denoted by the shaded area). Conservation of momentum implies that the momentum of the virtual photon is space-like. (b ) Consider an electron and a positron annihilating, e.g. in the c.m., into a (virtual) photon. The momentum of the virtual photon is time-like. (c ) Consider the scattering of an electron and a photon to a (virtual) electron. The momentum p of the virtual electron is off the mass-shell satisfying the relation p 2 < −m 2. (d ) Consider an electron becoming virtual in the scattering of the electron with the emission of a photon. The momentum p of the virtual electron is off the mass shell satisfying the relation p 2 > −m 2. The photon is denoted by a wavy line, while the electron (positron) by a solid one. A virtual particle which has too much energy or not enough energy to be on the mass shell, as the case may be, may, respectively, give off energy to another particle or absorb energy from another one and may eventually emerge as a real particle in an underlying Feynman diagram description of fundamental processes

    Full size image
  26. 26.

    The momentum conserving delta function occurs as follows. Invoking translational invariance, and the Hermiticity of the total momentum P which may equally operate to right or left in 〈p ′1 ν 1′, p ′2 ν 2′, … | P | p 1 ν 1, p 2 ν 2〉, leading to the two equal expressions: ( p 1 + p 2)〈p ′1 ν 1′, p ′2 ν 2′, … | p 1 ν 1, p 2 ν 2〉 = ( p ′1 + …)〈p ′1 ν 1′, p ′2 ν 2′, … | p 1 ν 1, p 2 ν 2〉. Hence upon subtracting one expression from the other, we obtain \(\big(\,p_{1} + p_{2} - (\,p\,'_{1}+\ldots )\big)\langle p\,'_{1}\nu _{1}',p\,'_{2}\nu _{2}',\ldots \vert p_{1}\nu _{1},p_{2}\nu _{2}\rangle = 0\). On the other the relation x f(x) = 0 implies that f(x) is proportional to δ(x).

  27. 27.

    This expression involves also the process e + e +  → e + e +.

  28. 28.

    For the process e + e +  → e + e +, only the term involving the pair η ​2(x) and \(\overline{\eta }_{1}(x)\) will contribute.

  29. 29.

    Also note that D μ ν(k 2 − k ′2) = D μ ν(k 1 − k ′1), and the symmetry of D μ ν in (μ, ν).

  30. 30.

    On the other hand, if you fix the final momenta and spins first, then the argument is reversed.

  31. 31.

    This property is usually attributed to Fermi, and follows from the formal equality δ (4)( p) |  p = 0 = ∫(dx)∕(2π)4 = VT∕(2π)4.

  32. 32.

    Note the sign change in the exponent of the following two equations.

  33. 33.

    This subsection is based on Manoukian and Yongram [61]. For the significance of these polarizations correlations in their role in support of the monumental quantum theory against so-called hidden variables theories, see, e.g., the just mentioned paper, Manoukian [56], Bell [5], and Bell and Aspect [6]. For a detailed review, with applications to QED, the electroweak theory and strings, see Yongram and Manoukian [83]. We have observed the speed dependence of polarization correlations in actual quantum field theory calculations early (2003) (see [82])

  34. 34.

    See also Problem 5.9.

  35. 35.

    For a simple structure we have \(\int (\mathrm{d}x)\,\overline{\eta }(x)\,S_{+}(x - x\,\:\!')\,\eta (x\,\:\!')(\mathrm{d}x\,\:\!') = \overline{\eta }\,\,S_{+}\eta\), in matrix multiplication form.

  36. 36.

    See Appendix III, at the end of the book, for dimensional regularization.

  37. 37.

    See also Sect. 4.1 for details on the concept of wavefunction renormalization.

  38. 38.

    This is due to the logarithmic nature on the cut-off dependence in QED. On the other hand, for the mass of the Higgs Boson, in the electroweak theory, one has a quadratic dependence on the cut-off Λ, and its unrenormalized and renormalized masses are drastically different.

  39. 39.

    In the next chapter, the gluon propagator, the counterpart of the photon propagator in QCD, will be directly analyzed with dimensional regularization.

  40. 40.

    The presence of − i ε in (k 2 + M 2 − i ε) in the denominator is understood.

  41. 41.

    A process involving no energy transfer.

  42. 42.

    See (3.2.22), (3.2.23), (3.2.24), (3.2.25) and (3.2.26).

  43. 43.

    Schwinger [68].

  44. 44.

    When the electron is treated non-relativistically, this correction has been also carefully computed (see Manoukian [56], p. 453 for a textbook treatment) to be: \(\big[1 +\kappa \, (\alpha /2\pi )\big]\), where κ = (16∕9) − 2 ln(3∕2) ≃ 0. 97, and compares well with the QED result.

  45. 45.

    See also (3.4.5), (3.4.6), (3.4.7), (3.4.8) and (3.4.9).

  46. 46.

    See, e.g., Yennie and Suura [81]. See also Manoukian [52].

  47. 47.

    Kusch and Foley [39], cf., Foley and Kusch [25].

  48. 48.

    See Lamb and Retherford [40] for the early experiment.

  49. 49.

    Sometimes I have the feeling that there are as many derivations of the Lamb shift as there are field theory practitioners.

  50. 50.

    A careful derivation of the Lamb shift was particularly carried out by Erickson and Yennie [22], see also Fox and Yennie [26]. There are also other careful derivations in the literature- too many to mention. A highly sophisticated and a very careful derivation, spelling out the finest of details, is also given by Schwinger [71].

  51. 51.

    For a treatment and a pedagogical presentation of the Lamb shift with the electron treated non-relativistically, see Manoukian [56], pp. 391–403, which is in the spirit of Bethe’s [8] classic treatment.

  52. 52.

    Note that m j  = m ℓ + m s , and for m s  = +1∕2:  m j  = +1∕2, m j  = −1∕2, and we have, respectively, m ℓ  = 0, −1. Similarly, for m s  = −1∕2:  we have m ℓ  = +1, 0.

  53. 53.

    For details on constructions of wavefunctions corresponding to the addition of spin and orbital angular momentum see, e.g., Manoukian [56], pp. 383, 384, 407.

  54. 54.

    A careful demonstration that this term is zero for ℓ = 0, \((\boldsymbol{\mathrm{L}}^{2} = 0)\), can be found on pp. 381–382 in Manoukian [56] in the analysis of the atomic fine-structure.

  55. 55.

    Here one may use the identity \(\boldsymbol{\mathrm{S}} \cdot \boldsymbol{\mathrm{ L}} =\big (\boldsymbol{\mathrm{J}}^{2} -\boldsymbol{\mathrm{ L}}^{2} -\boldsymbol{\mathrm{ S}}^{2}\big)/2\), and the fact that \(\big(j(j + 1) -\ell (\ell+1) - 3/4\big)/2 = -1\) for j = 1∕2, ℓ = 1.

  56. 56.

    Although this common constant is infinite, it may be defined with a cut-off.

  57. 57.

    This simply gives rise to a mass renormalization, see Manoukian [56], pp. 401–43. Also note that the matrix element 〈φ n  | [ . ] | φ n 〉, in (5.13.38) for \(H_{\:\!\text{C}} -\mathcal{E}_{n} \rightarrow 0\), simply factors out as 〈φ n  | φ n 〉 [ . ] = [ . ], giving an expression independent of the Coulomb functions and its related quantum numbers.

  58. 58.

    See, in particular, Bethe et al. [9] and Schwartz and Tiemann [67].

  59. 59.

    See, e.g., Kinoshita and Yennie [37], p. 7.

  60. 60.

    It is important to realize that the component A 0 is not a dynamical variable, i.e., its equation does not involve its time derivative, as is derived directly from the Lagrangian and need not, a priori assumed.

  61. 61.

    Note that the canonical conjugate momentum of A a is not ∂ 0 A a, for a = 1, 2.

  62. 62.

    This section is based on Manoukian [48]. For the relativistic invariance of QED in the Coulomb gauge, see Manoukian [51]. For the gauge invariance of transition amplitudes, see Manoukian [53].

  63. 63.

    Recall the notation Π σ δ(J σ) ≡ Π σ Π x  δ(J σ(x)).

  64. 64.

    Don’t let such an infinite proportionality constant scare you. We always divide by proportionality constants in 〈  ​0+∣0−〉 when determining Green functions.

  65. 65.

    See Problem 5.19.

  66. 66.

    This is the Fourier transform of the longitudinal part of the free photon propagator D μ ν, in the momentum description, with cut-offs, written as λ k μ k ν Λ 2∕[k 2(k 2 +μ 2)(k 2 +Λ 2)].

  67. 67.

    For the gauge invariance of transition amplitudes see Manoukian [53].

  68. 68.

    The derivation follows very closely to the ones in Problems 5.21 and 5.22.

  69. 69.

    Recall that 〈A μ(x)〉 \(\equiv (-\mathrm{i})\big[\updelta \,\text{F}/\updelta J_{\mu }(x)\big]/\text{F}\).

  70. 70.

    Takahashi [78]. The identity in differential form, that is for zero momentum transfer of the vertex function, is due to Ward [79]. Takahashi generalized it to non-zero momentum transfer.

  71. 71.

    See Appendix IV, below (IV.19), at the end of the book.

  72. 72.

    Functional methods in deriving equations for propagators and vertex functions were particularly used over the years by Schwinger and by his former students, notably by K. Johnson (see, e.g., [34]), and by many others.

  73. 73.

    The details of the derivation of the spectral representation may be omitted in a first reading.

  74. 74.

    The spectral representation of a propagator, in general, is usually attributed to Källen [35] and Lehmann [42], and is referred to as a Källen-Lehmann representation.

  75. 75.

    〈 . 〉 = 〈  ​0+ | . ∣0−〉∕〈  ​0+∣0−〉, with now all the external sources set equal to zero.

  76. 76.

    See (5.16.36).

  77. 77.

    Note that (Q 2 + M 2 − i ε) in the denominator, means that − Q 2 and M 2 may be interchanged, in a function of Q 2 or M 2 multiplying 1∕(Q 2 + M 2 − i ε), when applying the residue theorem for x 0 ≶ x  ​′ 0. Also note that the integrands in (5.16.50), (5.16.51) are even function in the variable Q μ. See also Brown [14], pp. 461–463.

  78. 78.

    A process involving no energy transfer.

  79. 79.

    See also the QED lowest contribution to ρ(M 2) in (5.10.55).

  80. 80.

    The minus sign is introduced to be consistent with the notation of the lowest order.

  81. 81.

    One may also equivalently subtract a Taylor expansion in Q, of the integrand in (5.16.93), in the process, in finally evaluating Π μ ν(Q).

  82. 82.

    See also Sect. 4.1.

  83. 83.

    See also the Introductory chapter of the book and Sect. 4.1.

  84. 84.

    See Appendix IV at the end of the book.

  85. 85.

    This is the content of a power counting theorem, due to Weinberg [80], which states that a graph is finite if all of its corresponding subintegrations, including the overall integration, are finite. For a pedagogical treatment of this technical problem see the author’s book on renormalization [45].

  86. 86.

    Finiteness is understood to refer to ultraviolet finiteness. The infrared problem is treated by the inclusion of soft photons in computing transition amplitudes. In the study of propagators and the vertex function, a non-zero photon mass is included in the analysis, as done earlier to the lowest order, to avoid infrared divergences in intermediate steps.

  87. 87.

    Light-light scattering subgraphs, as noted before, are convergent.

  88. 88.

    Finiteness of renormalized QED by induction was carefully treated in Bjorken and Drell [11], which was also applied to the renormalized photon propagator by a very elaborate method. We have preferred to treat the photon propagator by a method developed in a remarkable paper by Baker and Lee [4].

  89. 89.

    Note from (5.18.14), that to order n, d ren depends on (α π  c ) to order  < n and on the nth order as well, and the latter has been also established to be finite.

  90. 90.

    The renormalization group was introduced by Stueckelberg and Peterman [74], who also coined the name “renormalization group”. This was followed by other important work by Gell-Mann and Low [27], Ovsyannikov [65], Bogoliubov and Shirkov [12] and by many others.

  91. 91.

    See, e.g., Beringer et al. [7], Mele [63], and Erler [23].

  92. 92.

    I remember when I was a graduate student, I used to wonder, together with my fellow students, as to why nature chooses the value 1∕137 rather than some other value. Now with the rapid development of quantum field theory, together with higher energy experiments, we understand this value to be a reflection of the energies “at which we were then”. Effectively, this coupling changes as we move to higher energies attaining different numerical values as will be investigated in the next subsection. Although understanding why Nature chooses the low energy numerical value 1∕137 is still important, it does not seem to be as mysterious now as it was then.

  93. 93.

    In carrying the differentiation of this equation with respect to m, we follow the elegant approach, and to some extent the notation, of Adler [1].

  94. 94.

    Since π ​(Q 2) depends on the mass m through m 0, we may write: m(d∕dm) π ​(Q 2) = (m∕m 0) (d ​m 0∕dm) m 0 (∂ π (Q 2)∕∂ m 0).

  95. 95.

    Such a scaling equation is referred to as a Callan-Symanzik scaling equation [15, 75–77] .

  96. 96.

    This is fully justified by the application of a theorem due to Weinberg [80] when it is applied to the renormalized theory order by order [45], and essentially reduces to a dimensional analysis. The m 2∕Q 2 factor is already seen in (5.19.13) to lowest order.

  97. 97.

    de Rafaël and Rosner [18] and Baker and Johnson [3]. See also Gorishnii et al. [30]. Note that we use Adler’s β(α) function notation which is β(α)∕α of de Rafaël’s and Rosner’s.

  98. 98.

    The solution may be also written as: \(-y + \widetilde{G} \:\![\:\!\tau +y\:\!] =\tau -(\tau +y) + \widetilde{G} \:\![\:\!\tau +y\:\!] =\tau +G\:\![\:\!\tau +y\:\!]\) where \(\widetilde{G} \:\![\:\!\tau +y\:\!] - (\tau +y) = G\:\![\:\!\tau +y\:\!]\). Hence (5.19.38) is the general solution of (5.19.37).

  99. 99.

    See Adler [1], and, e.g., Manoukian [44].

  100. 100.

    One may similarly carry out an analysis of the wavefunction renormalization constant Z 2 and of the unrenormalized mass m 0 (see, e.g., [1]).

  101. 101.

    See, e.g., Krasnikov [38].

  102. 102.

    This is a consequence of the decoupling theorem and one carries out the analysis as if heavy masses in comparison to the energy of concern are not part of the dynamics. For intricacies, conditions and a proof of the decoupling theorem, see Manoukian [46, 47, 49] and other references therein.

  103. 103.

    See below (6.16.22) for experimental value.

  104. 104.

    Note that functional derivatives have now become simply partial derivatives.

  105. 105.

    Goldstone [28] and Goldstone et al. [29].

  106. 106.

    We will neglect radiative corrections in the subsequent analysis. Although we are not considering radiative corrections here, the λ(Φ † Φ)2 term in the Lagrangian density is important for renormalizability of the theory.

  107. 107.

    Note that if the coefficient of μ 2 is of opposite sign to the one in (5.20.26), one obtains v = 0.

  108. 108.

    Since we are not considering renormalizability aspects here, we use the notation for e.

  109. 109.

    Englert and Brout [21] and Higgs [31–33]. See also Anderson [2].

  110. 110.

    For the relevant details see Coleman and Weinberg [17].

  111. 111.

    This was introduced by Bialinicki-Birula and Bialinicka-Birula [10].

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    Edouard B. Manoukian

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Manoukian, E.B. (2016). Abelian Gauge Theories. In: Quantum Field Theory I. Graduate Texts in Physics. Springer, Cham. https://doi.org/10.1007/978-3-319-30939-2_5

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