Universit`a degli Studi di Padova
Degree in Physics
QFT
Notes written by master students on some of the topics
covered during the Quantum Field Theory course
Supervised by
Prof. Marco Matone
July 16, 2019
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Universit`a degli Studi di Padova
Degree in Physics
QFT
Notes written by master students on some of the topics
covered during the Quantum Field Theory course
Supervised by Prof. Marco Matone
July 16, 2019 Temporary Version
Corrections or suggestions can be reported to
Davide Dal Cin: davide.dalcin Marco Rigobello: marco.rigobello. Luca Teodori: luca.teodori Matteo Turco: matteo.turco. Marco Zecchinato: marco.zecchinato.
mail domain is studenti.unipd.it
Course Program
(i) 26/02/2018, 14.30 - 16.15. Aim and scope of the course. Description of the program and main references. Differences between Quantum Mechanics and Quantum Field Theories (QFT). Outlines on the perturbative formulation, operator formalism, path integral for- malism. Comments on φ^4 and Quantum Electrodynamics (QED). Triviality of φ^4 in D = 4 and its non-existence for D > 4. Non-Borel summability of QED. Wigner’s theorem and exact symmetries. Algebra of the observables and their *-automorphisms. The von-Neumann theorem and unitary equivalence of theo- ries with finitely many degrees of freedom. Spontaneous symmetry breaking as a phenomenon in theories with infinitely many degrees of freedom. Formulation of QFT on the lattice. Axiomatic approach. Wightman’s axioms and Wight- man’s reconstruction theorem. Euclidean formulation. Schwinger functions and Osterwalder-Schrader’s reconstruction theorem. The excellent book by F. Strocchi, “Elements of QM of infinite systems”, SISSA, Worlds Scientific, 1985, includes the analysis of some keys non-perturbative phe- nomena of QFT’s. Spontaneous symmetry breaking, shortly illustrated during the lecture, is reported at pp. 115-120. Other useful references are http: // arxiv. org/ pdf/ 1201. 5459. pdf , http: // arxiv. org/ pdf/ 1502. 06540. pdf and the two excellent books F. Strocchi, “An introduction to non-perturbative foundations of quantum field theory”, Oxford, 2013, Haag, “Local quantum physics, fields, par- ticles, algebras”, Springer-Verlag, 1996. See also the notes: A brief introduction to different QFT approaches. (ii) 01/03/2018, 13.30 - 15.15. Review of the Dirac equation. Gamma’s algebra and its representations. Covari- ance of the Dirac equation. Transformation of the spinors under the Poincar´e group, ψ′(x′) = S(Λ)ψ(x). Representations of the Poincar´e group. S(Λ) in the case of parity transformations. Charge conjugation. Itzykson-Zuber, sect. 2-1-2, 2-1-3, 2-4-2. Further references are chapters 2 and
v
vi Course Program
3 of Peskin-Schroeder, “Quantum Field Theory”, and the chapters 11, 12 and 13 of Bjorken-Drell, “Relativistic Quantum Fields”. Notes: Unitary representation of the Poincar´e Group - Wigner classification. Behaviour of local fields under the Poincar´e group. Relativistic covariance. Finite-dimensional irreducible represen- tations of the Lorentz group.
(iii) 08/03/2018, 13.30 - 15.15.
Clifford Algebra and bilinear spinors. S(Λ) in the case of charge conjugation. Discrete symmetries in the case of quantised fields. Parity operator. Itzykson-Zuber, sect. 3-4-1.
(iv) 12/03/2018, 14.30 - 16.15.
Charge conjugation and time reversal. Transformation properties of the Dirac bilinears under P , C and T transformations. P CT theorem. Itzykson-Zuber, sect. 3-4-2, 3-4-3, 3-4-4. See also chapter 2 and 3 of Peskin- Schroeder, “Quantum Field Theory”, and the chapters 11, 12 and 13 of Bjorken- Drell, “Relativistic Quantum Fields”. (v) 15/03/2018, 13.30 - 15.15. Lehmann, Symanzik and Zimmerman reduction formula. Dirac formulation of the path integral. Sect. 5 of M. Srednicki, “Quantum Field Theory”, Cambridge. See also the notes: K¨allen-Lehmann representation. N.B.: Srednicki uses the metric g′^ = −g, diag(g′) = (− 1 , 1 , 1 , 1). The scalar products defined by the two metrics have op- posite sign. Ramond, sect. 2.1 e 2.2. Notes: On the Dirac paper where it has been first formulated the path integral. P. A. M. Dirac, “The Lagrangian in quantum mechanics”, Phys. Z. Sowjetunion 3 (1933) 64. Reprinted in, Se- lected papers on quantum electrodynamics, J. Schwinger Ed., Dover, 1958. See also, http: // arxiv. org/ pdf/ quant-ph/ 0004090v1. pdf. The standard text for the path integral is Feynman-Hibbs, “Quantum mechanics and path integrals”, McGraw Hill, 1965, and the 2010 Dover edition commented by Styer.
(vi) 19/03/2018, 14.30 - 16.15.
Dirac paper on the formulation of the path integral. The need of the Hamiltonian formulation: the case H = p^2 v(q)/2. Ramond, sect. 2.2.
viii Course Program
(xii) 12/04/2018, 13.30 - 15.15. Relation between the connected two-point function and Γ(2). Determinants and heat equation. Effective action at order ℏ. Dependence of the coupling constant on the mass scale (Coleman-Weinberg). Ramond, sect. 3.4 and 3.5. Notes: ˜Γ(2) E (p) G˜(2) cE (p) = 1. Comments on Γ[ϕ] at order ℏ.
(xiii) 19/04/2018, 13.30 - 15.15.
Breaking of dilatation symmetry by quantum effects. Perturbation theory and Feynman rules. Examples of Feynman diagrams: tadpole, setting sun, fish. Normal ordering singularities. Ramond, sect. 3.6 and 4.1. (xiv) 23/04/2018, 14.30 - 16.15. Loop expansion as power expansion in ℏ. Truncated Green functions and LSZ re- duction formula. Superficial degree of divergence. Renormalisable, super-renormalisable and non-renormalisable theories. Weinberg theorem. Effective action as generating functional of proper vertices, Jona-Lasinio theorem. Ramond, sect. 4.2. Notes: Loop expansion as power expansion in ℏ. Truncated green functions and LSZ reduction formula. Effective action as generating func- tional of proper vertices, Jona-Lasinio theorem. The discussion at pp. 111-112 of the Ramond book is also reported, with more care and clarity, in the Casalbuoni lectures: pp. 92-97 of http: // theory. fi. infn. it/ casalbuoni/ dott1. pdf. See also pp. 139-142 of http: // theory. fi. infn. it/ casalbuoni/ lezioni99. pdf and sect. 11.5 of Kleinert book, “Particles and Quantum Fields”, World Sci- entific, 2016. (xv) 26/04/2018, 13.30 - 15.15. Jona-Lasinio theorem. Regularisation methods. Dimensional regularisation. Notes: Effective action as generating functional of proper vertices, Jona-Lasinio theorem. Ramond, sect. 4.3.
Course Program ix
(xvi) 03/05/2018, 13.30 - 15. Dimensional regularisation. Proof of the Feynamn parametrisation formula. Eu- clidean action in 2ω dimension, dimensionless λnew and ’t Hooft mass parameter μ. Tadpole and fish diagrams. Ramond, sect. 4.3. The proof of the Feynman parametrisation formula follows the one in http: // kodu. ut. ee/ ~ kkannike/ english/ science/ physics/ notes/ feynman_ param. pdf. (xvii) 07/05/2018, 14.30 - 16.15. Calculation of the fish and double scoop diagrams. Notes on the calculation of the setting sun diagram, analysis of the divergences, residue depending on the moment. Renormalisation. On the Feynman rules. Mass term considered as a two-leg vertex, Feynman propagator with mass as diagrammatic series of the massless one with interaction given by the mass term. Counterterms for Γ˜(2)(p) and Γ˜(4)(p). Recursive structure of the renormalisation procedure. Ramond, sect. 4.4 and 4.5. Notes: On the Feynman rules. ˜Γ(2)(p) at one-loop with the counterterm contribution. A useful reference for further information is section 11 of the Kleinert text, “Particles and Quantum Fields”, World Scientific,
(xviii) 10/05/2018, 13.30 - 15.
Renormalised Lagrangian density. Relation between the bare and renormalised proper vertex functions. Renormalisation group equation. Scale equation for ˜Γ(renN^ ). Bare parameters in terms of λ, m/μ ed . Renormalisation prescriptions. ’t Hooft and Weinberg prescriptions. The β function. Landau pole. Ultraviolet and infrared fixed points of β. Asymptotic freedom and confinement. Scaling of Γ˜(renN^ ) and anomalous dimension. Ramond, sect. 4.5 and 4.6. Notes: Relation between the bare and renormalised proper vertex functions. Scaling of Γ˜(renN^ ) and anomalous dimension. The explicit steps concerning the equations 4.6.10 - 4.6.15 of the Ramond book are reported in the equations 31.11 - 31.23 of http: // theory. fi. infn. it/ casalbuoni/ dott1. pdf. The english version is reported in Chapter 6 of http: // theory. fi. infn. it/ casalbuoni/ lezioni99. pdf. (xix) 14/05/2018, 14.30 - 16.15. Calculation of γm and γd. Vertex functions in the limit of large momenta in the case of a UV fixed point. Prescription dependence of the renormalisation group coefficients. Prescription independence of the existence of a UV fixed point of the
Course Program xi
(xxiv) 31/05/2018, 13.30 - 15.
Counterterms of QED at one loop. Renormalisation at one-loop. Lamb Shift. Anomalous magnetic moment. L.H. Ryder, “Quantum Field Theory”, sect. 9.5 and 9.6. Itzykson-Zuber, sect. 2-2-3.
xii Course Program
xiv Main References
Chapter 1
Overview of the formulations of
Quantum Field Theory
1
The aim of these chapter is to provide a short overview on the various approaches to quantum field theory (QFT), whose main task is to compute physical quantities such as the S -matrix and therefore the cross section of the theory.
We will start with the axiomatic approach, based on the Wightman axioms, which is mathematically well-defined and is therefore used for rigorous proofs. Another approach is the perturbative one, which is the most used for studying quantum field theories. This can be formulated in terms of the operator approach or in the framework of the path integral formalism. A non-perturbative approach to QFT concerns the formulation on a lattice, where space-time is discretised. We will also shortly review the formalism based on the Schr¨odinger representation of quantum fields. The last section concerns a short introduction to the phenomenon of spontaneous symmetry breaking. Here is the list of acronyms used in these notes.
CCR canonical commutation relations GF Green’s functions QCD quantum chromodynamics QED quantum electrodynamics QFT quantum field theory QFTL quantum field theory on a lattice QM quantum mechanics SSB spontaneous symmetry breaking
(^1) Matteo Sighinolfi
1
Overview of the formulations of Quantum Field Theory 3
this means that φ(t, x) satisfies the free wave equation
1 c^2
∂^2
∂t^2 φ(t,^ x)^ − ∇
(^2) φ(t, x) + m (^2) φ(t, x) = 0. (1.1)
It is now possible to choose as units of time and space x^0 = ct , xj^ , j = 1, 2 , 3. In this way, the Minkowski metric is the familiar mostly negative
g ≡ gμν = diag(1, − 1 , − 1 , −1).
Adopting the standard convention for covariant and contravariant variables, the free wave equation is now written in the Lorentz covariant form
∂μ∂μφ + m^2 φ = 0. (1.2)
This equation can be obtained from the free action
S 0 =^12
d^4 x
∂μφ∂μφ + m^2 φ^2
To have an interacting theory one adds a term to S 0 which is usually a polynomial in φ with grade higher than two, for example
SI =
d^4 x λ 4 φ^4 ,
implying the classical equation of motion
∂μ∂μφ + m^2 φ − λφ^3 = 0. (1.4)
Until now there is nothing new or tricky in our physics, but by now things starts getting more difficult. If φ(t, x) is a real field, then Eq.(1.4) have smooth solutions for any smooth bounded initial conditions at some initial time t 0. The field is determined at every position and time knowing its value and its time derivative at t = t 0. At any time, there is a Poisson bracket between the field and its time derivative φ˙ { φ(t, x), φ˙(t, y)
= δ(3)(x − y).
If one tries to quantise the field φ, it is clear that it cannot be a function of x because of the above Poisson bracket containing δ(3)(x − y), which is a distribution. The only possibility for φ is to be a distribution in the sense of Schwartz. Looking back at Eq.(1.4) we see that the term φ^3 is problematic because non-linear distributions are undefined. Actually, while quantising the theory one unavoidably gets the divergences in the calculations, like the infinities arising in the Dyson-Feynman theory.
4 Chapter 1
A different approach was successfully implemented by Wightman in 1956 for free fields. Wightman found that to give sense to the space-time derivatives of the free field, and also to field polynomials and their derivatives, it is enough to smear the field with an infinitely smooth function of Schwartz class S(R^4 ) in space-time.^3 In particular, Wightman showed that the smeared field
φ( l k)(f ) =
d^4 x φ( l k)(x)f (x) , (1.5)
with f (x) a test function and
φ( l k)(f )
linear operators in a Hilbert space H, is a well- defined operator on the Fock space. The main problem with the Wightman axioms, is that all known four-dimensional theories satisfying Wightman’s axioms have a trivial scattering matrix. Nevertheless, non-trivial theories satisfying the Wightman axioms exist in lower dimension.
1.2.1 Wightman’s axioms
It is now necessary to introduce a set of axioms to work with our QFT, where the fields are the smeared ones in (1.5).
W1 (Relative invariance of the space of states). It exists a Hilbert space H that carries a continuous unitary representation U(Λ, a) of the Poincar´e spinorial group (universal covering group of the Poincar´e proper group).
W2 (Spectral properties). The spectrum of pμ^ is concentrated exclusively in the superior closed cone V +^ :=
p ∈ M | p^2 ≥ 0 , p^0 ≥ 0
m = 0 included.
W3 (Existence and uniqueness of the vacuum). ∃! a vacuum state | 0 〉 (up to a phase eiα) for H that is invariant under U(Λ, a).
With these three axioms Wightman noticed that for the quantised field φ, φ(f ) is unbounded. For an unbounded operator it is necessary to define a domain D
W4 (Fields’ domain of definition). The components φ( l k)of the field φ(k)^ are operators with distributional values on the Schwartz’s space S(M), with domains of definition D
(^3) The space S := S(R (^4) ) consists of infinitely differentiable real functions of real variables that goes to zero at infinity faster than any power of the Euclidean distance. For an introduction to distributions see, for example, [12].