Skip to main content
SpringerLink
Log in

Formalism for QFT

  • Chapter
  • First Online:
Entanglement Measures and Their Properties in Quantum Field Theory

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 34))

  • 841 Accesses

Abstract

This chapter presents the most important notions and examples of the theory of operator algebras. These are then used to formulate the basic principles of quantum field theory and some examples of algebraic QFTs.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 49.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 64.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    In general, a v. Neumann algebra is isomorphic to a v. Neumann algebra in standard form if it has a faithful representation which in turn is the case if it has a faithful normal state, i.e. a normal state such that \(\omega (a^*a) = 0\) implies \(a=0\). In the following, we will always assume that this is the case.

  2. 2.

    It is understood here that b acts on \( \langle \psi |\) in \(\bar{\mathbb {C}}^N\) by \(b \langle \psi | \equiv \langle b^* \psi |\).

  3. 3.

    As a vector space, the graded tensor product is first defined to be the usual (algebraic) tensor product. The product is defined as \((a_1 \hat{\otimes }b_1)(a_2 \hat{\otimes }b_2)= (-1)^{deg(a_2)deg(b_1)} \, a_1 a_2 \hat{\otimes }b_1 b_2\) and the *-operation is \((a \hat{\otimes }b)^* = (-1)^{deg(a)deg(b)} a^* \hat{\otimes }\ b^*\), where the degree is defined to be 0 resp. 1 for even resp. odd elements under \(\alpha \). It is then shown that a natural \(C^*\)-norm compatible with these relations and the above isomorphism can be defined which extends the \(C^*\)-norm of \({\mathfrak C}(K_i, \Gamma _i)\). The graded tensor product is the \(C^*\)-closure under this norm.

  4. 4.

    The covering group is needed to describe non-integer spin.

  5. 5.

    In terms of algebras:

    $$ \pi _0(\mathfrak {A}) = \left( \bigcup _{x \in \mathbb {R}^{d,1}} \pi _0(\mathfrak {A}(O+x)) \right) '' $$

    for any causal diamond O.

  6. 6.

    If there are none, then the norm is set to infinity.

  7. 7.

    Here the braces denote “generated by, as a \(C^*\)-algebra”, and \(\mathrm{supp}(F)=\mathrm{supp}(q)\cup \mathrm{supp}(p)\), where the support \(\mathrm{supp}\) of a function is the closure of the set of all points where it does not vanish.

  8. 8.

    Our convention for the Fourier transform in one dimension is \(\widetilde{f}(p) = \frac{1}{\sqrt{2\pi }} \int \mathrm{d}x f(x) e^{-ipx}\).

  9. 9.

    In our setup, we arrive at the normalization

    $$\begin{aligned}{}[a(\mathbf{k}), a^\dagger (\mathbf{k}')] = 2\omega (\mathbf{k}) \ \delta ^d(\mathbf{k}-\mathbf{k}') \cdot 1 \ , \quad [a(\mathbf{k}), a(\mathbf{k}')] = 0 . \end{aligned}$$
    (2.37)
  10. 10.

    The complex conjugate, \(\bar{V}\), of a vector space V is identical as a set, but has the scalar multiplication \(\lambda \cdot v \equiv \bar{\lambda } v\).

  11. 11.

    Apart from the value m of the mass of the basic particle.

  12. 12.

    I.e., it satisfies the relations of the permutation group.

  13. 13.

    Informally, \(z^\dagger (\theta ) = z(\theta )^*\).

  14. 14.

    J turns out to be equal to the modular conjugation associated with the algebra \(\mathfrak {R}\) defined below.

  15. 15.

    Note that [35] contained an error, which has been amended in [40]. At present, the arguments only establish (a1)–(a4), (a5’) for regions O of a minimal size, contrary to the claim of [35].

  16. 16.

    One sometimes requires that the symmetry algebra of the net is the full Virasoro algebra, i.e. that the net contains the algebra of quantized diffeomorphisms as a subnet. Then the split property is automatic [43].

References

  1. R.V. Kadison, J.R. Ringrose, Fundamentals of the Theory of Operator Algebras (Academic Press, New York, I 1983, II 1986)

    Google Scholar 

  2. O. Bratteli, D.W. Robinson, Operator Algebras and Quantum Statistical Mechanics (Springer, I 1987, II 1997)

    Google Scholar 

  3. R.T. Powers, E. Størmer, Free states of the canonical anticommutation relations. Commun. Math. Phys. 16, 1–33 (1970)

    Article  ADS  MathSciNet  Google Scholar 

  4. H. Araki, Relative entropy for states of von Neumann algebras. Publ. RIMS Kyoto Univ. 11, 809–833 (1976)

    Article  MathSciNet  Google Scholar 

  5. H. Araki, Relative entropy for states of von Neumann algebras II. Publ. RIMS Kyoto Univ. 13, 173–192 (1977)

    Article  MathSciNet  Google Scholar 

  6. M. Florig, S.J. Summers, On the statistical independence of algebras of observables. J. Math. Phys. 38, 1318 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  7. S. Doplicher, R. Longo, Standard and split inclusions of von Neumann algebras. Invent. Math. 75, 493 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  8. D. Buchholz, E.H. Wichmann, Causal independence and the energy level density of states in local quantum field theory. Commun. Math. Phys. 106, 321 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  9. E. Binz, R. Honegger, A. Rieckers, Construction and uniqueness of the \(C^*\)-Weyl algebra over a general pre-symplectic space. J. Math. Phys. 45, 2885–2907 (2004)

    Article  ADS  Google Scholar 

  10. F. Kärsten, Report MATH, 89–06 (Akademie der Wissenschaften der DDR, Karl-Weierstrass-Institut für Mathematik, Berlin, 1989)

    Google Scholar 

  11. R.M. Wald, Quantum Field Theory in Curved Space-Time and Black Hole Thermodynamics (Chicago University Press, Chicago, 1994)

    MATH  Google Scholar 

  12. J. Manuceau, A. Verbeure, Quasi-free states of the C.C.R.-algebra and Bogoliubov transformations. Commun. Math. Phys. 9, 293–302 (1968)

    Article  ADS  MathSciNet  Google Scholar 

  13. H. Araki, S. Yamagami, On quasi-equivalence of quasifree states of the canonical commutation relations. Publ. RIMS Kyoto Univ. 18, 283–338 (1982)

    MATH  MathSciNet  Google Scholar 

  14. P. Leylands, J.E. Roberts, D. Testard, Duality for Quantum Free Fields (Preprint CNRS, Marseille, 1978)

    Google Scholar 

  15. H. Araki, On quasifree states of the CAR and Bogoliubov automorphisms. Publ. RIMS Kyoto Univ. 6, 385–442 (1970)

    Article  MathSciNet  Google Scholar 

  16. J. Cuntz, Simple C*-algebras generated by isometries. Commun. Math. Phys. 57, 173–185 (1977)

    Article  ADS  MathSciNet  Google Scholar 

  17. S. Hollands, R.M. Wald, Quantum fields in curved spacetime. Phys. Rept. 574, 1 (2015)

    Article  ADS  MathSciNet  Google Scholar 

  18. H. Araki, Mathematical Theory of Quantum Fields (Oxford Science Publications, 1993)

    Google Scholar 

  19. H. Reeh, S. Schlieder, Bemerkungen zur Unitäräquivalenz von lorentzinvarianten Feldern. Nuovo Cimento 22, 1051–1068 (1961)

    Article  Google Scholar 

  20. R. Haag, J.A. Swieca, When does a quantum field theory describe particles? Commun. Math. Phys. 1, 308–320 (1965)

    Article  ADS  MathSciNet  Google Scholar 

  21. A. Pietsch, Nuclear Locally Convex Spaces (Springer, Berlin, 1972)

    Book  Google Scholar 

  22. C.J. Fewster, I. Ojima, M. Porrmann, p-nuclearity in a new perspective. Lett. Math. Phys. 73, 1–15 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  23. D. Buchholz, K. Fredenhagen, C. D’Antoni, The universal structure of local algebras. Commun. Math. Phys. 111, 123 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  24. C.J. Fewster, R. Verch, The necessity of the Hadamard condition. Class. Quant. Grav. 30, 235027 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  25. D. Buchholz, C. D’Antoni, R. Longo, Nuclear maps and modular structures. 1. General properties. J. Funct. Anal. 88, 223 (1990)

    Article  MathSciNet  Google Scholar 

  26. G. Lechner, K. Sanders, Modular nuclearity: a generally covariant perspective. Axioms 5, 5 (2016)

    Article  Google Scholar 

  27. D. Buchholz, P. Jacobi, On the nuclearity condition for massless fields. Lett. Math. Phys. 13, 313 (1987)

    Article  ADS  MathSciNet  Google Scholar 

  28. C. Bär, N. Ginoux, F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics (European Mathematical Society Publishing House, Germany, 2007)

    Book  Google Scholar 

  29. R. Seeley, Complex powers of elliptic operators. Proc. Symp. Pure Appl. Math. AMS 10, 288–307 (1967)

    Article  MathSciNet  Google Scholar 

  30. B.S. Kay, Linear spin-zero quantum fields in external gravitational and scalar fields I. A one particle structure for the stationary case. Commun. Math. Phys. 62, 55–70 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  31. M. Reed, B. Simon, Methods of Modern Mathematical Physics II (Academic Press, 1975)

    Google Scholar 

  32. C. D’Antoni, S. Hollands, Nuclearity, local quasiequivalence and split property for Dirac quantum fields in curved space-time. Commun. Math. Phys. 261, 133 (2006)

    Article  ADS  Google Scholar 

  33. J. Figueroa-O’Farrill, Majorana Spinors, http://www.maths.ed.ac.uk/jmf/Teaching/Lectures/Majorana.pdf

  34. B. Schroer, H.W. Wiesbrock, Modular constructions of quantum field theories with interactions. Rev. Math. Phys. 12, 301 (2000)

    Article  MathSciNet  Google Scholar 

  35. G. Lechner, Construction of quantum field theories with factorizing s-matrices. Commun. Math. Phys. 277, 821 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  36. G. Lechner, On the existence of local observables in theories with a factorizing S-matrix. J. Phys. A 38, 3045–3056 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  37. D. Buchholz, G. Lechner, Modular nuclearity and localization. Ann. Henri Poincaré 5, 1065 (2004)

    Article  ADS  MathSciNet  Google Scholar 

  38. R. Haag, Local Quantum Physics: Fields, Particles, Algebras (Springer, Berlin, 1992)

    Book  Google Scholar 

  39. E. Abdalla, C. Abdalla, K.D. Rothe, Non-perturbative Methods in 2-Dimensional Quantum Field Theory (World Scientific, New Jersey, 1991)

    Book  Google Scholar 

  40. S. Alazzawi, G. Lechner, Inverse Scattering and Locality in Integrable Quantum Field Theories, arXiv:1608.02359 [math-ph]

  41. K. Fredenhagen, J. Hertel, Local algebras of observables and point-like localized fields. Commun. Math. Phys. 80, 555 (1981)

    Article  ADS  Google Scholar 

  42. H. Bostelmann, Phase space properties and the short distance structure in quantum field theory. J. Math. Phys. 4, 052301 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  43. V. Morinelli, Y. Tanimoto, M. Weiner, Conformal Covariance and the Split Property, Commun. Math. Phys. 357(1), 379-406 (2018)

    Article  ADS  MathSciNet  Google Scholar 

  44. Y. Kawahigashi, Conformal field theory, tensor categories and operator algebras. J. Phys. A 48 (30), 303001, 57 (2015)

    Article  MathSciNet  Google Scholar 

  45. K.-H. Rehren, Algebraic conformal quantum field theory in perspective, in Advances in Algebraic Quantum Field Theory (Springer, Switzerland, 2015), pp. 331–364. Math. Phys. Stud

    Chapter  Google Scholar 

  46. Y. Kawahigashi, R. Longo, Classification of local conformal nets: case c < 1. Ann. Math. 160, 493 (2004)

    Article  MathSciNet  Google Scholar 

  47. P. DiFrancesco, P. Mathieu, D. Senechal, Conformal Field Theory (Springer, New York, 1997)

    Book  Google Scholar 

  48. C.J. Fewster, S. Hollands, Quantum energy inequalities in two-dimensional conformal field theory. Rev. Math. Phys. 17, 577 (2005)

    Article  MathSciNet  Google Scholar 

  49. V.T. Laredo, Integrating unitary representations of infinite-dimensional Lie groups. J. Funct. Anal. 161, 478–508 (1999)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute for Theoretical Physics, University of Leipzig, Leipzig, Sachsen, Germany

    Stefan Hollands

  2. School of Mathematical Sciences, Dublin City University, Dublin, Ireland

    Ko Sanders

Authors
  1. Stefan Hollands
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ko Sanders
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ko Sanders .

Rights and permissions

Reprints and permissions

Copyright information

© 2018 The Author(s), under exclusive licence to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Hollands, S., Sanders, K. (2018). Formalism for QFT. In: Entanglement Measures and Their Properties in Quantum Field Theory. SpringerBriefs in Mathematical Physics, vol 34. Springer, Cham. https://doi.org/10.1007/978-3-319-94902-4_2

Download citation

Publish with us

Policies and ethics

Search

Navigation