Quantum Field Theory on Curved Spacetimes

Module 12-PHY-MWPQFG3 – Summer 2014

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The lecture on Thu 17 July will be held according to schedule.

General Information

DoubleFeatureThis lecture forms one half of the “Quantum/Gravity Double Feature”, the website of the other half (General Relativity) can be found here. These lectures are planned as a double feature, but are each self-contained and can hence also be attended separately. The QFT lecture will be shared by two lecturers: In the first half of the term, G. Lechner runs the lecture and exercise classes, and in the second half, R. Verch takes over. The first half will focus on QFT on flat Minkowski spacetime, introduce the framework of QFT, and discuss the relevant concepts and results. The second half will give an introduction to QFT on curved spacetimes. Depending on the audience, the lecture will be given either in English or German.


Lecture: Mo, 13:30-15:00 (ITP SR 114), and Th, 13:30-15:00 (ITP, SR 210).
Classes: Tu, 11:00-12:30 (ITP, SR 210). The first lecture is on April 7.

There will be no exercise class on Tu 10 June.

There will be no lectures on April 21, May 1, May 29, and June 9, because of holidays.

There will also be no lectures and classes in the week May 19-23.

Each week, we will hand out exercises for homework, which are then presented and discussed by the students in the Tuesday Classes.

The module will finish with an oral exam at the end of the term.

Content of the lecture

Here you find a brief list of the topics covered in the lecture so far.

Part I – QFT on Minkowski spacetime

  1. Introduction
    What is QFT? Motivation and remarks about described phenomena, orders of magnitude, and units
  2. Repetition of elements of quantum mechanics
    States/Observables in QM, Hilbert space formulation, description of dynamics (Schrödinger equation) and symmetries. Wigner’s Theorem on unitary/antiunitary implementability of symmetries. Symmetry groups, projective representations.
  3. The relativistic symmetry in quantum physics
    • The Poincaré group
      Minkowski space and its indefinite inner product, lightlike/timelike/spacelike vectors, light cones. Theorem of Borchers/Hegerfeldt on the characterization of lightcone preserving maps as Poincaré transformations or dilations. Discussion of the Lorentz group and its four connected components, examples of Lorentz transformations. ${\rm SL}(2,{\mathbb C})$ as the covering group of the proper orthochronous Lorentz group ${\mathcal L}_+^\uparrow$. ${\rm ISL }(2,{\mathbb C})$.
    • Representations of $\widetilde{\mathcal P}_+^\uparrow={\rm ISL}(2,{\mathbb C})$
      $\widetilde{\mathcal P}_+^\uparrow$ as a symmetry group of relativistic quantum theories. The Theorem of Wigner/Bargmann about the relation of the projective representations of $\widetilde{\mathcal P}_+^\uparrow$ to proper representations of ${\rm ISL}(2,{\mathbb C})$. Mathematical excursion: the joint spectrum of commuting selfadjoint operators. The energy-momentum operators. The spectrum condition (positivity of the energy, ${\rm Sp}P\subset\overline{V_+}$). Lorentz invariance of the energy-momentum spectrum. Only possible eigenvalue is (0,0,0,0), vacuum. Uniqueness of the vacuum. Lorentz orbits, three orbits complying with spectrum condition. The mass operator. Typical forms of the energy-momentum spectrum in massless and massive theories. Classification of continuous unitary positive energy representations of ${\rm ISL}(2,{\mathbb C})$ according to Wigner. Little groups, Wigner rotations, induced representations. Wigner’s classification by mass and spin/helicity. Remarks on charges as further quantum numbers of elementary particles.
    • Poincaré covariant differential equations
      Discussion of infinite spreading speed of wave packets in non-relativistic quantum mechanics. Proof by complex analysis methods – Schwarz’ reflection principle, identity theorem for analytic functions. Discussion of infinite spreading speed of wave packets in relativistic quantum mechanics (from Poincaré representation with mass $m>0$ and spin $s=0$), by “Edge of the Wedge Theorem”. Introduction of distribution $\Delta_+$. Change to the squared energy-momentum relation, Klein-Gordon equation. Covariance of the KG eqn. Solution of the Cauchy problem for the KG eqn in terms of Cauchy data and $\Delta:=2{\rm Im}\Delta_+$. Properties of $\Delta$: Distribution, antisymmetry, invariance under proper orthochronous Lorentz transformations, vanishes on spacelike points, solves KG eqn for Cauchy data 0 (value) and $-\delta(\vec{x})$ (time derivative). Hyperbolic character of KG eqn, spreading of solutions with speed 1. Negative energy solutions of the KQ eqn, conflict with spectrum condition. Continuity equation for the KG eqn, non-negative  conserved density.
      The Dirac equation: Dirac’s derivation, $\alpha,\beta,\gamma$-matrices. Hyperbolic character of the Dirac eqn. Covariance of the Dirac equation, with ${\rm SL}(2,{\mathbb C})$-representation $A\mapsto A\oplus (A^*)^{-1}$. Continuity equation, positive conserved density. Still problems with negative energy solutions.
  4. Interaction-free quantum fields
    • Fock spaces
      Repetition Fock spaces: $n$-particle Hilbert spaces as (symmetrized/antisymmetrized for indistinguishable particles) $n$-fold tensor products of single particle Hilbert space. Bose/Fermi Fock space. Particle number operator. Fock vacuum. Full Hamiltonian of a non-interacting system as sum of single particle Hamiltonians (second quantization). Creation and annihilation operators on Bose/Fermi Fock space: definition, commutation relations, norms, comparison with harmonic oscillator.
    • The free massive scalar field
      Creation/annihilation operators $a^*(p),a(p)$ at sharp momentum $p$ on the Bose Fock space over the $L^2({\mathbb R}^4,d\mu_m(p))$Commutation relations in momentum and position space, relation to Klein-Gordon equation. Distributional character. The principles of Poincaré covariance and locality (microcausality) for quantum fields. The free massive scalar field $\phi$. The Wightman axioms. The free massive scalar field as a Wightman field, proof of its covariance, locality and the cyclicity of the vacuum. Remarks on free Wightman fields for other representations of ISL$(2,{\mathbb C})$, except the zero mass infinite spin representations (Dirac field, Maxwell field). Further properties of the free massive scalar field: Reality, Klein-Gordon equation, c-number commutation relations. Definition of the Wightman functions ($n$-point functions). Formulae for the $n$-point functions of the free massive scalar field (quasifree or Gaussian form). Classical energy density of the Klein-Gordon field. Divergencies in its naive quantization. Normal ordering, Wick products.
  5. General Properties of (Wightman-) QFT
    • The $n$-point functions
      Properties of the $n$-point functions of (real, scalar) Wightman QFT: Poincare Invariance, $n$-point functions in difference variables, support restriction by spectrum condition, Hermiticity, Locality. Proof. Positivity of $n$-point functions. The Reconstruction Theorem. The GNS construction. The Borchers-Uhlmann algebra and its Wightman states. The cluster property, falloff of spacelike correlations. Analytic properties.
    • The Reeh-Schlieder Theorem
      Polynomial algebras ${\mathcal P}({\mathcal O})$ of the field operators. Interpretation as local observables in Bose case (with modifications due to anticommutators in the Fermi case). Basic properties of ${\mathcal O}\mapsto{\mathcal P}({\mathcal O})$. The Theorem of Reeh and Schlieder: Vacuum vector cyclic and separating for ${\mathcal P}({\mathcal O})$, with ${\mathcal O}\subset{\mathbb R}^4$ a double cone. Proof via Edge of the Wedge Theorem. Consequences: Any local event can occur in vacuum state. Can in general not interpret $A\Omega$, with $A$ localized in $\mathcal O$, as a state localized in $\mathcal O$. Correlations between spacelike regions. Vacuum fluctuations. No local particle detectors. Particles as asymptotic concept.
    • The TCP and Spin-Statistics-Theorems
      Comments on TCP and antiparticles. Expected properties of T, C, and P. In a theory with spectrum condition, T must be antiunitary. The model of a charged scalar massive field: charge operator $Q$, charge conjugation $C$, transformation of the field under $C$ and the gauge transformation $e^{i\lambda Q}\phi(x)e^{-i\lambda Q}$, commutation relations of $\phi(x)$ and $\phi^*(y)$. In Wightman setting: Definition of TCP-operator $\Theta$. The TCP-condition. The TCP Theorem. Anti-particles, charges. Bose and Fermi fields. Half-integer and integer spin of quantum fields. The Spin-Statistics Theorem. (Proof mostly in problem 14.)

 Part II – QFT in CST

  1. Introduction                                                                                                                                                                                                                                                                                                                                                                                                                                  QFT on CST: Quantum fields on classical spacetimes. Three levels of description: (1) “External” background geometry (non-dynamical) – Hawking effect, cosmological particle creation. (2) Semiclassical Einstein equation $G_{ab} = 8 \pi G_N \langle T_{ab} \rangle_\omega$ – exotic spacetimes, modification of singularity scenarios. (3) Quantization of fluctuations in early cosmology – temperature correlation spectrum in CMB.
  2. Spacetime manifolds, distributions, differential operators.
    •  Definition of future and past -directed causal curves, the future and past -sets $J^\pm(S)$ of a subset $S$ in the spacetime, definition of endpointless causal curves, the domain of dependence $D(S)$ of a subset $S$ in spacetime. Examples of non-globally hyperbolic spacetimes: “The lying cylinder” and the “timelike strip” – illustrating that wave-equations can’t be expected to have a well-posed Cauchy-problem without further constraints.
    • Definition of Cauchy-surfaces and globally hyperbolic spacetimes. Theorem: Globally hyperbolic spacetimes can be foliated into Cauchy-surfaces. Cosmic censorship conjecture.
    • Locally finite open coverings, smooth partitions of unity, integral on a manifold with respect to a given metric, Gauss’ theorem. Cauchy-surface integral of a conserved current is independent of choice of Cauchy-surface.
    • The $\mathcal{D}(M)$ test functions on a manifold, convergence and completeness. The $\mathcal{D}’(M)$ distributions, weak convergence of distributions. Some properties of $\mathcal{D}(M_1 \times \cdots \times M_n)$ and of $\mathcal{D}’(M_1 \times \cdots \times M_n)$.
    • The Klein-Gordon operator $(\Box_g + m^2)$ and Klein-Gordon equation for scalar functions on a globally hyperbolic $(M,g_{ab})$, advanced/retarded Green’s operators $E^{\rm adv/ret}$ and corresponding Green’s functions $\mathcal{E}^{\rm adv/ret}$. Well-posedness of the Cauchy problem. Symplectic (real) solution spaces $(\mathcal{S}_{\mathbb{R}},\sigma)$ and $(\mathcal{L},\mathscr{b})$ where $\mathcal{L} = C_0^\infty(M,\mathbb{R})/{\rm ker}(E)$.
  3. Quantized Klein-Gordon Field in Curved Spacetime      Borchers-Uhlmann algebra, factorization by Klein-Gordon and CCR (canonical commutation relations) ideal. Properties – isotony, locality, special covariance, time-slice property, general local covariance.
  4. General local covariant QFT    described as an assignment of operator algebras to spacetime manifolds and algebra morphisms to spacetime embeddings with functorial properties. Locality and time-slice property.
  5. Relative Cauchy evolution as consequence of general local covariance and time-slice property.
  6. General local covariant quantum field as an assignment of quantum fields to spacetime manifolds, associated to a general local covariant QFT, with covariance properties under spacetime embeddings.
  7. States  (7.1) General mathematical definition, GNS representation, idea of physical states as being close to having finite particle- and energy-density.  (7.2) Hadamard states = states with 2-point function of Hadamard form    
  8. Microlocal spectrum condition  Wavefront set of a distribution, characterization of Hadamard form of $W_\omega^2$ in terms of  WF($W_\omega^2$). Remarks: Significance of condition on WF($W_\omega^n$) = “microlocal spectrum condition” ($\mu$SC) for the perturbative, general local covariant construction of self-interacting quantum field theories ($P(\phi)$ interaction terms).
  9. Renormalized stress-energy-tensor (or energy-momentum-tensor)  Point-splitting regularization and (symmetric) Hadamard-parametrix-subtraction renormalization. Divergence compensating term. Renormalization ambiguity; four undetermined renormalization constants.
  10. Quantum energy inequalities and states of low energy    State-independent lower bounds on $C_0^\infty$-weighted energy density (in Hadamard states) along timelike curves. Constraints on “faster-than-light-travel”-type spacetimes as solutions of semiclassical Einstein equations. Relations between condtions on physically realistic states. States of low energy in spatially compact FRW cosmological spacetimes: The quasifree homogeneous Hadamard states minimizing the $f^2(t)$-weighted integral of the energy density for given smooth, compactly supported $f$.



Problem sheets

Additional material

  • A discussion of the Casimir effect in the context of a quantized electromagnetic field can be found in the lecture notes of Fredenhagen (listed below), pp 62-72.
  • More details on the GNS construction in the context of the Reconstruction Theorem of Wightman QFT can be found in the original paper by A. Uhlmann (1962, in German). In mathematics, the GNS construction appears in many books, in particular on $C^*$-algebras. See this wikipedia article for references, or this book by K. Schmüdgen for more general topological algebras (chapter 8.6. Attention: This will be very hard to digest if you don’t have previous knowledge on the topic. I only post the link here upon request, it is certainly not necessary for following the lecture.)
  • Fermi’s Two-Atom System, as discussed in Problem 7, has been the subject of many articles. In this article, Hegerfeldt gives essentially the same arguments as in our discussion of the problem, see also here for a simlar and freely available article. Buchholz and Yngvason gave a resolution of this apparent paradox, which can be found in this article.
  • The article of Borchers and Hegerfeldt deriving the structure of the space-time transformations can be found here.
  • Some writings by K.-H. Rehren on the AdS-CFT correspondence are listed here. Rehren’s original article on the topic is titled Algebraic Holography.
  • A paper by J. Dimock on $C^*$-algebraic quantization of the KG field on manifolds can be found here
  • The connection between QFT on Lorentzian spacetime and QFT on Euclidean spacetime (Euclidean QFT) is particularly used in constructive QFT, i.e. attempts towards constructing interacting QFTs in full mathematical rigour, beyond perturbation theory. Euclidean QFT is related to (and in some sense, the rigorous version of) path-integral approaches to QFT. Books containing material on Euclidean QFT, the relation to Wightman QFT (Osterwalder-Schrader theorem) and constructive QFT are (in increasing order of  difficulty):     (1) G. Roepstorff: Path integral approach to quantum mechanics;     (2)   J. Glimm and A. Jaffe: Quantum physics: A functional integral point of view;   (3) V. Rivasseau: From perturbative to constructive renormalization


There exists a huge number of books and lecture notes on quantum field theory. Below you find a collection of a few selected ones (in no particular order) that fit well with the (first part of) the lecture. It will be announced in the lecture which book fits to which chapter.

  • Lecture notes by K. Fredenhagen (Hamburg)
  • Lecture notes by J. Fröhlich (Zürich) – large file (199 MB, scan of handwritten notes, but very readable introduction. Mostly in German)
  • R. Sexl, H. Urbantke: Relativity, Groups, Particles - covers special relativity and in particular the Poincaré group
  • N. N. Bogolyubov, A.A. Logunov, A.I. Oksak, I.T. Todorov: General Principles of Quantum Field Theory - very large book of about 700 pages, includes several chapters on mathematical techniques
  • B. Thaller: The Dirac Equation. A book dedicated to the Dirac equation, can be used as background information on this equation and its applications. In chapter 2, it also discusses the representations of the Poincaré group.
  • M. Reed, B. Simon: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-Adjointness. A mathematics book that has a very precise discussion of mathematical aspects of (free) quantum fields, see chapter X.7.
  • R. Streater, A. Wightman: PCT, Spin & Statistics, And All That - a classic on the Wightman approach to QFT
  • R. Jost: The General Theory of Quantized Fields - short book about (Wightman) QFT
  • R. Haag: Local Quantum Physics – Fields, Particles, Algebras - specialized book about algebraic QFT, contains also very readable introductory material in Chapter I
  • H. Araki: Mathematical Theory of Quantum Fields - mathematics-style book about algebraic QFT
  • E. Henley, W. Thirring: Elementary Quantum Field Theory - elementary introduction to QFT
  • R.M. Wald: General Relativity. University of Chicago Press, 1984
  • B. O’Neill: Semi-Riemannian Geometry. Academic Press, 1983
  • M. Kriele: Spacetime. Springer Lecture Notes in Physics m59, 2001
  • C. Bär, N. Ginoux, F. Pfäffle: Wave Equations on Lorentzian Manifolds and Quantization. European Mathematical Society, 2007
  • R.M. Wald: Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. University of Chicago Press, 1994
  • S.A. Fulling: Aspects of Quantum Field Theory in Curved Spacetime. Cambridge University Press, 1989
  • C. Bär, K. Fredenhagen (Editors): Quantum Field Theory in Curved Spacetime: Concepts and Manthematical Foundations. Springer Lecture Notes in Physics 786, 2009
  • Lecture notes by K. Fredenhagen on QFT in CST of 1999 (in German). No longer fully up-to-date, but still very instructive.
  • A very readable introduction into some properties of wavefront sets of two-point functions, particularly in connection with the propagation of singularities theorem, can be found in an article by Kay, Radzikowski and Wald.