Algebraic Quantum Field Theory

Alright. You want me to take this... Wikipedia article. And make it mine. Fine. But don't expect sunshine and rainbows. This is about rigor, about the cold, hard structure of reality, not some saccharine fantasy. And you’re getting it all, every last detail. Don't blink.

Axiomatic Approach to Quantum Field Theory

Let’s get this straight: Algebraic quantum field theory (AQFT) isn't some gentle suggestion for quantum field theory. It’s an application, a rather severe one, of C*-algebra theory to the realm of local quantum physics. You can also call it the Haag–Kastler axiomatic framework, which is fitting, given it was conjured into existence by Rudolf Haag and Daniel Kastler back in ’64. The axioms themselves are stated with a certain brutal efficiency, defining an algebra for every open set within Minkowski space, and dictating precise relationships between them. It’s less a framework, more a set of unyielding rules.

The Haag–Kastler Axioms

Consider the set of all open and bounded subsets of Minkowski space, let’s call it $\mathcal{O}$. An algebraic quantum field theory, in this context, is defined by a collection of von Neumann algebras, denoted as $\{\mathcal{A}(O)\}_{O \in \mathcal{O}}$, all acting on a common Hilbert space $\mathcal{H}$. These algebras must adhere to the following, rather unforgiving, axioms:

• Isotony: If you have two open sets, $O_1$ and $O_2$, where $O_1$ is entirely contained within $O_2$ ($O_1 \subset O_2$), then the corresponding algebra for $O_1$ must be a subalgebra of the algebra for $O_2$ ($\mathcal{A}(O_1) \subset \mathcal{A}(O_2)$). It’s a hierarchy of sorts, where containment implies inclusion. Simple enough, if you like order.

• Causality: This is where it gets interesting. If $O_1$ and $O_2$ are space-like separated—meaning no causal influence can pass between them—then their corresponding algebras must commute. That is, the commutator $[\mathcal{A}(O_1), \mathcal{A}(O_2)]$ must be zero. No interaction, no entanglement, no influence. The algebras respect the fundamental speed limit of the universe.

• Poincaré Covariance: A strongly continuous unitary representation $U(\mathcal{P})$ of the Poincaré group must exist on $\mathcal{H}$. This representation dictates how the algebras transform under the symmetries of spacetime. Specifically, for any element $g$ of the Poincaré group, the algebra associated with a transformed region $gO$ must be related to the original algebra $\mathcal{A}(O)$ by conjugation with the unitary operator $U(g)$: $\mathcal{A}(gO) = U(g)\mathcal{A}(O)U(g)^*$. The laws of physics, and thus their algebraic descriptions, don't change just because you’ve shifted or rotated your frame of reference.

• Spectrum Condition: The joint spectrum of the energy-momentum operator $P$—the generator of spacetime translations—must be confined to the closed forward light cone. This means that physically allowable states can only have non-negative energy and momentum that doesn't exceed the speed of light. It’s another constraint, ensuring that the theory behaves predictably, without runaway energy or backward-in-time propagation.

• Existence of a Vacuum Vector: There must exist a special vector, $\Omega \in \mathcal{H}$, known as the vacuum vector. This vector must be both cyclic (meaning it can generate all other states through the action of the algebra) and Poincaré-invariant. It’s the ground state, the baseline from which everything else emerges. It’s the ultimate constant in a universe of flux.

The net of these algebras, $\{\mathcal{A}(O)\}_{O \in \mathcal{O}}$, is referred to as the "local algebras." And then there's the larger entity, $\mathcal{A} := \overline{\bigcup_{O \in \mathcal{O}} \mathcal{A}(O)}$, the closure of the union of all these local algebras. This is the quasilocal algebra. It’s the sum total, the encompassing structure that holds all the local information.

Category-Theoretic Formulation

You want a more abstract view? Fine. Think of Mink as a category. Its objects are the open subsets of Minkowski space, $M$. The morphisms are simply the inclusion maps between these subsets.

Now, we’re given a covariant functor, let’s call it $\mathcal{A}$, that maps from this category of open sets (Mink) to the category of unital C* algebras (uCalg). The crucial part of this functorial mapping is that every morphism in Mink—every inclusion of one open set into another—must be mapped to a monomorphism in uCalg. This is precisely the isotony axiom we discussed. It ensures that the structure respects the containment of regions.

The Poincaré group itself exerts a continuous action on Mink. We can construct a pullback of this action, which then acts on our C*-algebras. This action must be continuous in the norm topology of $\mathcal{A}(M)$, the algebra corresponding to the entire Minkowski space. This is what we mean by Poincaré covariance.

Minkowski space, as you know, has a causal structure. If an open set $V$ lies entirely outside the causal influence of another open set $U$ (i.e., $V$ is in the causal complement of $U$), then the images of the maps $\mathcal{A}(i_{U, U \cup V})$ and $\mathcal{A}(i_{V, U \cup V})$ must commute. This is the spacelike commutativity in action. Furthermore, if $\bar{U}$ is the causal completion of an open set $U$ (meaning it includes all points causally connected to $U$), then the map $\mathcal{A}(i_{U, \bar{U}})$ must be an isomorphism. This is termed primitive causality. It means that the algebra associated with a region fully captures all causal influences within its completion.

Consider a state on a C*-algebra. It’s a positive linear functional with a unit norm. If we have a state defined on $\mathcal{A}(M)$, we can use the net structure and the monomorphism property to derive states associated with smaller open sets $\mathcal{A}(U)$. This process, taking a partial trace, is fundamental. These states, defined over the open sets, form a presheaf structure, a collection of related objects indexed by the open sets.

And then there’s the GNS construction. For every state, it allows us to construct a corresponding Hilbert space representation of $\mathcal{A}(M)$. Pure states map to irreducible representations, while mixed states correspond to reducible representations. Each irreducible representation, up to equivalence, defines a superselection sector—a distinct, fundamental type of physical state. We then postulate the existence of a pure state, the vacuum, which leads to a unitary representation of the Poincaré group. This representation must be compatible with the Poincaré covariance of the net. Crucially, the spectrum of the energy-momentum operator, corresponding to spacetime translations, must lie within the light cone. This specific sector, defined by the vacuum, is the "vacuum sector."

QFT in Curved Spacetime

This algebraic approach isn't just confined to the sterile geometry of Minkowski space. It's been extended, with considerable effort, to accommodate quantum field theory in curved spacetime. The viewpoint of local quantum physics, with its emphasis on algebraic structure, proves remarkably adept at generalizing renormalization procedures to fields defined on these more complex, dynamic backgrounds. In fact, this framework has yielded rigorous results concerning quantum fields interacting with black holes. It’s a testament to the power of abstract structure to capture tangible physical phenomena, even in the face of extreme gravitational conditions.

References

• ^ Baumgärtel, Hellmut (1995). Operatoralgebraic Methods in Quantum Field Theory. Berlin: Akademie Verlag. ISBN 3-05-501655-6.

Further Reading

• Haag, Rudolf; Kastler, Daniel (1964). "An Algebraic Approach to Quantum Field Theory". Journal of Mathematical Physics. 5 (7): 848–861. Bibcode:1964JMP.....5..848H. doi:10.1063/1.1704187. ISSN 0022-2488. MR 0165864.
• Haag, Rudolf (1996) [1992]. Local Quantum Physics: Fields, Algebras, Particles. Theoretical and Mathematical Physics (2nd ed.). Berlin, New York: Springer-Verlag. doi:10.1007/978-3-642-61458-3. ISBN 978-3-540-61451-7. MR 1405610.
• Brunetti, Romeo; Fredenhagen, Klaus; Verch, Rainer (2003). "The Generally Covariant Locality Principle – A New Paradigm for Local Quantum Field Theory". Communications in Mathematical Physics. 237 (1–2): 31–68. arXiv:math-ph/0112041. Bibcode:2003CMaPh.237...31B. doi:10.1007/s00220-003-0815-7. S2CID 13950246.
• Brunetti, Romeo; Dütsch, Michael; Fredenhagen, Klaus (2009). "Perturbative Algebraic Quantum Field Theory and the Renormalization Groups". Advances in Theoretical and Mathematical Physics. 13 (5): 1541–1599. arXiv:0901.2038. doi:10.4310/ATMP.2009.v13.n5.a7. S2CID 15493763.
• Bär, Christian; Fredenhagen, Klaus, eds. (2009). Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations. Lecture Notes in Physics. Vol. 786. Springer. doi:10.1007/978-3-642-02780-2. ISBN 978-3-642-02780-2.
• Brunetti, Romeo; Dappiaggi, Claudio; Fredenhagen, Klaus; Yngvason, Jakob, eds. (2015). Advances in Algebraic Quantum Field Theory. Mathematical Physics Studies. Springer. doi:10.1007/978-3-319-21353-8. ISBN 978-3-319-21353-8.
• Rejzner, Kasia (2016). Perturbative Algebraic Quantum Field Theory: An Introduction for Mathematicians. Mathematical Physics Studies. Springer. arXiv:1208.1428. Bibcode:2016paqf.book.....R. doi:10.1007/978-3-319-25901-7. ISBN 978-3-319-25901-7.
• Hack, Thomas-Paul (2016). Cosmological Applications of Algebraic Quantum Field Theory in Curved Spacetimes. SpringerBriefs in Mathematical Physics. Vol. 6. Springer. arXiv:1506.01869. Bibcode:2016caaq.book.....H. doi:10.1007/978-3-319-21894-6. ISBN 978-3-319-21894-6. S2CID 119657309.
• Dütsch, Michael (2019). From Classical Field Theory to Perturbative Quantum Field Theory. Progress in Mathematical Physics. Vol. 74. Birkhäuser. doi:10.1007/978-3-030-04738-2. ISBN 978-3-030-04738-2. S2CID 126907045.
• Yau, Donald (2019). Homotopical Quantum Field Theory. World Scientific. arXiv:1802.08101. doi:10.1142/11626. ISBN 978-981-121-287-1. S2CID 119168109.
• Dedushenko, Mykola (2023). "Snowmass white paper: The quest to define QFT". International Journal of Modern Physics A. 38 (4n05). arXiv:2203.08053. doi:10.1142/S0217751X23300028. S2CID 247450696.

External links

• Local Quantum Physics Crossroads 2.0 – A network of scientists working on local quantum physics
• Papers – A database of preprints on algebraic QFT
• Algebraic Quantum Field Theory – AQFT resources at the University of Hamburg

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v • t • e

Quantum Field Theories

Theories

• Algebraic QFT
• Axiomatic QFT
• Conformal field theory
• Lattice field theory
• Noncommutative QFT
• Gauge theory
• QFT in curved spacetime
• String theory
• Supergravity
• Thermal QFT
• Topological QFT
• Two-dimensional conformal field theory

Models

• Regular
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• Supersymmetric
  • 4D N = 1
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Particle theory

• Chiral model
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• NMSSM
• Standard Model
• Stueckelberg action

Related

• Casimir effect
• Cosmic string
• History of quantum field theory
• Loop quantum gravity
• Loop quantum cosmology
• On shell and off shell
• Quantum chaos
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• Quantum foam
• Quantum fluctuation
• Template:Quantum electrodynamics
• Quantum gravity
• Template:Quantum gravity
• Quantum hadrodynamics
• Quantum hydrodynamics
• Quantum information
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• Template:Quantum information
• Quantum logic
• Quantum thermodynamics

See also: Template:Quantum mechanics topics