Fundamental Aspects of Quantum Field Theory

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.

Appendices

Appendix A: Basic Equalities Involving the CPT Operator

We have the following explicit equalities for the CPT operator Θ :

$$\displaystyle\begin{array}{rcl} & & \qquad \qquad \qquad \varTheta \,\Big[\overline{\eta }(x)\,\psi (x) + \overline{\psi }(x)\,\eta (x)\Big]\,\varTheta ^{-1} \\ & =\,& \Big[-\,(\gamma \,^{0})_{ ab}\,(\gamma \,^{5}\gamma \,^{0})_{ bc}\,\overline{\psi }_{c}(-x)\,\eta _{a}(x) +\,\eta _{ a}^{{\ast}}(x)\,\psi _{ b}(-x)(\gamma \,^{0}\gamma \,^{5})_{ ba}\Big] \\ & & \quad \, =\,\Big [(\gamma \,^{5})_{ ac}\,\overline{\psi }_{c}(-x)\,\eta _{a}(x) -\,\eta _{a}^{{\ast}}(x)\,\psi _{ b}(-x)(\gamma \,^{5}\gamma \,^{0})_{ ba}\Big] \\ & & \qquad \quad \;\; =\,\;\Big [\overline{\psi }(-x)\,\gamma \,^{5}\,\eta (x) -\,\overline{\eta }(x)\,\gamma \,^{5}\,\psi (-x)\Big]. {} \\ \end{array}$$

(A-4.1)

Equation (4.10.8) now follows upon changing the variable of integration x​ → ​− x. Equations (4.10.9), (4.10.10) are self evident.

Finally, to establish (4.10.7), we consider transformation properties of the product of fields and their derivatives that may appear in $\mathcal{L}(x)$. Consider first Dirac fields, and a general linear combination of, say, two Dirac fields

$$\displaystyle{ \frac{1} {2}\,M_{ab}^{\,\mu \,\!_{1}\ldots \mu \:\!_{k} }[\overline{\psi }_{a}^{1}(x),\psi \:\!_{ b}^{2}(x)], }$$

(A-4.2)

where $M_{ab}^{\mu _{1}\ldots \mu \:\!_{k}}$ is of the general form, or a linear combination of such tensors as, $\big(\:\!\gamma ^{\mu _{1}}\ldots \gamma ^{\mu \:\!_{k}}\big)$, $\big(\:\!\gamma ^{\mu _{1}}\ldots \gamma ^{\mu \:\!_{k}}\gamma \,^{5}\big)$. Partial derivatives will be considered in studying the general structure of tensor components in (A-4.5) below. In view of the application of a CPT transformation, note that

$$\displaystyle{ [\,(\gamma \,^{5}\gamma \,^{0})\gamma ^{\:\!\mu \,\!_{1}{\ast}}\ldots \gamma ^{\:\!\mu \:\!_{k}{\ast}}(\gamma \,^{0}\gamma \,^{5})\,]_{ ab} = (-1)^{k}\,[\,\gamma ^{\mu \,\!_{1}\!\top }\ldots \gamma ^{\:\!\mu \:\!_{k}\!\top }\,]_{ ab} = (-1)^{k}\,[\,\gamma ^{\mu \:\!_{k} }\ldots \gamma ^{\,\!\mu \,\!_{1} }\,]_{ba}\,, }$$

(A-4.3)

and similarly in the presence of an additional γ ⁵ matrix, where we have used the properties: γ ⁵ γ ^μ = −γ ^μ γ ⁵, (γ ⁰)² = (γ ⁵)² = 1, γ ⁰ γ ^μ∗ = γ ^μ​⊤ γ ⁰, γ ^0⊤ = γ ⁰, γ ^5⊤ = γ ⁵. Hence from the transformation rule of a spinor given in (4.10.2), we obtain

$$\displaystyle{ \varTheta \,\Big(\frac{1} {2}\,M_{ab}^{\:\!\mu \,\!_{1}\ldots \mu \:\!_{k} }[\,\overline{\psi }_{a}^{1}(x),\psi \,_{ b}^{2}(x)\,]\Big)\,\varTheta ^{-1}\Big\vert _{ x\rightarrow -x} = (-1)^{k}\,\Big(\frac{1} {2}\,M_{ab}^{\:\!\mu \,\!_{1}\ldots \mu \:\!_{k} }[\,\overline{\psi }_{a}^{1}(x),\psi \,_{ b}^{2}(x)\,]\Big)^{\dag }. }$$

(A-4.4)

Note the importance of using the anti-symmetric average (see Sect. 4.9) in establishing this result.

On the other hand for any tensor component $\chi _{\sigma \,\!_{ 1}\ldots \sigma \:\!_{k}}(x)$, such as constructed out of products of vector fields, of scalar fields and their derivatives, we obviously have under the CPT transformations (4.10.3) and (4.10.4),

$$\displaystyle{ \varTheta \,\chi _{\sigma \,\!_{1}\ldots \sigma \:\!_{k}}(x)\,\varTheta ^{-1}\, =\, (-1)^{k}\,\chi _{ \sigma \,\!_{1}\ldots \sigma \:\!_{k}}^{\dag }(-x). }$$

(A-4.5)

Upon defining

$$\displaystyle{ M_{ab}^{\mu \,\!_{1}\ldots \mu \:\!_{k} }\,\frac{1} {2}\,[\,\overline{\psi }\,\!_{a}^{1}(x),\psi \,_{ b}^{2}(x)\,]\, =\,\varOmega ^{\mu \,\!_{1}\ldots \mu \:\!_{k} }(x), }$$

(A-4.6)

we may infer from (A-4.4) and (A-4.5) that

$$\displaystyle{ \varTheta \,\frac{1} {2}\,\{\varOmega ^{\mu \,\!_{1}\ldots \mu \:\!_{k}}(x),\chi _{\sigma \,\!_{ 1}\ldots \sigma \:\!_{k}}(x)\}\,\varTheta ^{-1}\Big\vert _{ x\rightarrow -x}\, =\,\Big (\frac{1} {2}\,\{\varOmega ^{\mu \,\!_{1}\ldots \mu \:\!_{k}}(x),\chi _{\sigma \,\!_{ 1}\ldots \sigma \:\!_{k}}(x)\}\Big)^{\dag }. }$$

(A-4.7)

This establishes (4.10.7) since a Lagrangian density, must contain an even number of Lorentz indices and all products in it are symmetrized with respect to Bose-Einstein factors and anti-symmetrized with respect to Fermi-Dirac fields, and due to the Hermiticity of the Lagrangian density, and finally followed by a change of the variable of integration x​ → ​−x.

Problems

4.1. Use (4.2.23), (4.2.24), (4.2.25), (4.2.26), (4.2.27), (4.2.28), (4.2.29), (4.2.30) and (4.2.31), with the definition of the unit vector, $\boldsymbol{\mathrm{n}}$ as given in (4.2.24) and the explicit expression of the rotation matrix [ R ^{i j} ​ ] in (2.2.11) with ξ = θ to derive  (4.2.32).

4.2. Show that

$$\displaystyle\begin{array}{rcl} & & \qquad \qquad \;\;\;\;\;\text{exp}\,[\,-\mathrm{i}\alpha \:\!J\,\!_{03}\,]\,\text{exp}\,[\,-\mathrm{i}\theta \:\!\boldsymbol{\mathrm{n}} \cdot \boldsymbol{\mathrm{ J}}\,]\,W\,^{0} {}\\ & & =\, (W^{3}\:\!\sinh \,\alpha + W^{0}\:\!\cosh \,\alpha )\,\:\!\text{exp}\,[\,-\mathrm{i}\alpha \:\!J\,\!_{ 03}\,]\,\text{exp}\,[\,-\mathrm{i}\theta \:\!\boldsymbol{\mathrm{n}} \cdot \boldsymbol{\mathrm{ J}}\,], {}\\ \end{array}$$

thus verifying (4.2.34).

4.3. We are given two orthogonal four vectors P ^μ W  _μ  = 0, and P ^μ P  _μ  = 0, W ^μ W  _μ  = 0, $P\,^{0} = \vert \boldsymbol{\mathrm{P}}\vert > 0$. Show that these imply that, W ^μ = λ P ^μ for some numerical λ.

4.4. Derive the explicit expressions for $W\,^{0},\boldsymbol{\mathrm{W}}$ in (4.2.16).

4.5. Derive the transformation law in (4.3.48) as it follows from (4.3.47).

4.6. Under a Lorentz transformation: x → x −δx = x ′, δx ^μ = − δω ^{μ ν} ​x _ν , a vector field A _α (x) has the following transformation law¹⁹

$$\displaystyle{ A'_{\alpha }(x\,\:\!') =\Big (\delta _{\alpha }\:\!^{\beta } + \frac{\mathrm{i}} {2}\,\updelta \omega ^{\nu \lambda }\big(S_{\nu \lambda })_{\alpha }\:\!^{\beta }\Big)\,A_{\beta }(x), }$$

where note that the argument of A′ _α  on the left-hand-side is x  ​′ and not x, and the spin structure is given by

$$\displaystyle{ \big(S_{\nu \lambda }\big)_{\alpha }\:\!^{\beta } = \frac{1} {\mathrm{i}} \,\big(\eta _{\nu \:\!\alpha }\,\eta _{\lambda }\:\!^{\beta } -\eta _{\lambda \:\!\alpha }\,\eta _{\nu }\:\!^{\beta }\big). }$$

Given the Maxwell Lagrangian is: $\mathcal{L} = -(1/4)F^{\alpha \beta }F_{\alpha \beta }$ where F ^{α β} = (∂ ^α A ^β − ∂ ^β A ^α): (i) find the (symmetric) energy-momentum tensor, (ii) show it is gauge invariant, (iii) show directly that it is conserved. What is its trace?. [Note that the canonical energy momentum tensor is not gauge invariant.]

4.7. Use the formal expression of the Dirac Lagrangian density: $\mathcal{L} = -\overline{\psi }\big(\gamma \partial /\mathrm{i}) + m\big)\psi$, to show that the (symmetric) energy-momentum tensor is given by: $T^{\mu \nu } = (1/4\,\mathrm{i})\overline{\psi }\big(\gamma ^{\mu }\mathop{\partial }\limits^\longleftrightarrow ^{\nu } +\gamma ^{\nu }\mathop{\partial }\limits^\longleftrightarrow ^{\mu }\big)\psi$, where $\mathop{\partial }\limits^\longleftrightarrow ^{\nu } =\overrightarrow{ \partial }^{\nu } -\overleftarrow{ \partial }^{\nu }$.

4.8. Find the expressions of the components T ⁰⁰(x) T ^{0 ​ ​k}(x) of the energy-momentum tensor of the Dirac quantum field in Problem 4.7 involving no time derivatives.

4.9. Derive the following fundamental equal-time commutation relation of the component T ⁰⁰ of the energy-momentum tensor for the Dirac theory in Problems 4.7 and 4.8: [ T ⁰⁰(x), T ⁰⁰(x  ​′) ] $= -\mathrm{i}\,\big[\,T^{0\:\!k}(x) + T^{0\:\!k}(x\,\:\!')\,\big]\partial _{k}\delta ^{3}(\boldsymbol{\mathrm{x}} -\boldsymbol{\mathrm{ x}}\:\!')$. This was derived by Schwinger [16, 17]. Commutations relations of all the components of the energy-momentum tensor in the full QED theory, taking into account of the gauge problem, are derived in Manoukian [5].

4.10. Given three 3-vectors $\boldsymbol{\mathrm{e}}_{+},\,\boldsymbol{\mathrm{e}}_{-},\,\boldsymbol{\mathrm{p}}/\vert \boldsymbol{\mathrm{p}}\vert$, satisfying

$$\displaystyle{ \boldsymbol{\mathrm{e}}_{\pm }\cdot \,\boldsymbol{\mathrm{ e}}_{\mp }^{{\ast}} = 0,\quad \boldsymbol{\mathrm{e}}_{ \pm }\cdot \,\boldsymbol{\mathrm{ e}}_{\pm }^{{\ast}} = 1,\quad \frac{\boldsymbol{\mathrm{p}}} {\vert \boldsymbol{\mathrm{p}}\vert }\cdot \,\boldsymbol{\mathrm{ e}}_{\pm } = 0,\quad \delta ^{ij} = \frac{p\,^{i}} {\vert \boldsymbol{\mathrm{p}}\vert }\frac{p\,^{i}} {\vert \boldsymbol{\mathrm{p}}\vert } +\mathrm{ e}\,_{+}^{i{\ast}}\,\mathrm{e}\,_{ +}^{j} +\mathrm{ e}\,_{ -}^{i^{{\ast}} }\,\mathrm{e}\,_{-}^{j}, }$$

with the last relation providing a completeness relation in 3D space, verify explicitly the four dimensional completeness relation (4.7.129) in Minkowski spacetime.

4.11. Show that the equation

$$\displaystyle{ \Big[\eta ^{\mu \nu }\Big(\frac{\gamma \partial } {\mathrm{i}} + m\Big) -\frac{1} {3}\Big(\gamma \:\!^{\mu }\frac{\partial \:\!^{\nu }} {\mathrm{i}} +\gamma \:\! ^{\nu }\frac{\partial \:\!^{\mu }} {\mathrm{i}} \Big) + \frac{1} {3}\gamma ^{\mu }\Big(\!-\frac{\gamma \partial } {\mathrm{i}} + m\!\Big)\gamma ^{\nu }\Big]\psi _{\nu }(x)\! =\! K^{\mu }(x), }$$

also leads to the equations in (4.7.137), (4.7.138) for K ^μ = 0, and that it merely adds the non-propagating term − (2∕3 m ²)[ p ​^μ γ ​^ν − p ​^ν γ ​^μ + (γ p − m)γ ^μ γ ^ν ] to the propagator ρ ​^{μ ν}( p)∕[ p ​² + m ² − iε ]. This is the equation used, for example, by Takahashi [19].

4.12. Derive the constraint on the spatial components ψ ⁱ(x) of the m ≠ 0 Rarita-Schwinger field in (4.7.148).

4.13. Derive the basic Eq. (4.7.150) needed for the m ≠ 0 Rarita-Schwinger field components to satisfy the constraint in (4.7.148).

4.14. Establish the spin 3/2 nature of the m = 0 Rarita-Schwinger field in (4.7.193).

4.15. The propagator for a massive neutral vector field was derived in (4.7.106), (4.7.107), (4.7.108) and (4.7.109), and may be obtained by functional differentiation

$$\displaystyle{ (\mathrm{i})\,(-\mathrm{i}) \frac{\updelta } {\updelta K_{\mu }(x)}(-\mathrm{i}) \frac{\updelta } {\updelta K_{\nu }(x\,\:\!')}\langle \,\,\,\!0_{+}\!\mid 0_{-}\rangle \Big\vert _{K^{\sigma }=0} =\varDelta \:\!_{ +}^{\mu \nu }(x - x\,\:\!'), }$$

directly from the vacuum-to-vacuum transition amplitude 〈  ​0₊​∣0₋〉 given in (4.7.108). Use the identity in (4.6.38), following from the quantum dynamical principle, to show that the time ordered product of the vector fields $\langle \,\,\,\!0_{+}\!\mid \big(V ^{\mu }(x)V ^{\nu }(x\,\,\:\!')\big)_{+}0_{-}\rangle \Big\vert _{K^{\sigma }=0}$ satisfies

$$\displaystyle{ \mathrm{i}\,\langle \,\,\,\!0_{+}\!\mid \big(V ^{\mu }(x)V ^{\nu }(x\,\:\!')\big)_{+}0_{-}\rangle \Big\vert _{K^{\sigma }=0} =\varDelta _{+}\:\!^{\mu \nu }(x - x\,\:\!') - \frac{1} {m^{2}}\,\delta \:^{\nu }\:\!_{ 0}\eta \:^{0\:\mu }\,\delta ^{(4)}(x - x\,\:\!'). }$$

That is, (i × time-ordered-product) of the vector fields not only does not represent the propagator of the vector field, but is also not covariant.

4.16. Show that if the interaction Lagrangian density in (4.8.11) is replaced, say, by $\mathcal{L}_{I}(x) =\lambda \, (\gamma ^{\mu })_{ab}\,[\overline{\psi }_{a}(x),\psi _{a}(x)]\,V _{\mu }(x)/2$, which is linear in V _μ (x), then there is no modifying factor as in (4.8.29) in the action integral.