Axiomatic approach to quantum field theory
Algebraic quantum field theory (AQFT ) is an application to local quantum physics of
C*-algebra theory. Also referred to as the HaagâKastler
axiomatic framework for
quantum field theory , because it was introduced by
Rudolf Haag and
Daniel Kastler (
1964 ). The axioms are stated in terms of an algebra given for every open set in
Minkowski space , and mappings between those.
HaagâKastler axioms
Let
O
{\displaystyle {\mathcal {O}}}
be the set of all open and bounded subsets of Minkowski space. An algebraic quantum field theory is defined via a set
{
A
(
O
)
}
O
∈
O
{\displaystyle \{{\mathcal {A}}(O)\}_{O\in {\mathcal {O}}}}
of
von Neumann algebras
A
(
O
)
{\displaystyle {\mathcal {A}}(O)}
on a common
Hilbert space
H
{\displaystyle {\mathcal {H}}}
satisfying the following axioms:
[1]
Isotony :
O
1
⊂
O
2
{\displaystyle O_{1}\subset O_{2}}
implies
A
(
O
1
)
⊂
A
(
O
2
)
{\displaystyle {\mathcal {A}}(O_{1})\subset {\mathcal {A}}(O_{2})}
.
Causality : If
O
1
{\displaystyle O_{1}}
is space-like separated from
O
2
{\displaystyle O_{2}}
, then
A
(
O
1
)
,
A
(
O
2
)
=
0
{\displaystyle [{\mathcal {A}}(O_{1}),{\mathcal {A}}(O_{2})]=0}
.
Poincaré covariance : A strongly continuous unitary representation
U
(
P
)
{\displaystyle U({\mathcal {P}})}
of the Poincaré group
P
{\displaystyle {\mathcal {P}}}
on
H
{\displaystyle {\mathcal {H}}}
exists such that
A
(
g
O
)
=
U
(
g
)
A
(
O
)
U
(
g
)
∗
,
g
∈
P
.
{\displaystyle {\mathcal {A}}(gO)=U(g){\mathcal {A}}(O)U(g)^{*},\,\,g\in {\mathcal {P}}.}
Spectrum condition : The joint spectrum
S
p
(
P
)
{\displaystyle \mathrm {Sp} (P)}
of the energy-momentum operator
P
{\displaystyle P}
(i.e. the generator of space-time translations) is contained in the closed forward lightcone.
Existence of a vacuum vector : A cyclic and Poincaré-invariant vector
Ω
∈
H
{\displaystyle \Omega \in {\mathcal {H}}}
exists.
The net algebras
A
(
O
)
{\displaystyle {\mathcal {A}}(O)}
are called local algebras and the C* algebra
A
:=
⋃
O
∈
O
A
(
O
)
¯
{\displaystyle {\mathcal {A}}:={\overline {\bigcup _{O\in {\mathcal {O}}}{\mathcal {A}}(O)}}}
is called the quasilocal algebra .
Category-theoretic formulation
Let Mink be the
category of
open subsets of Minkowski space M with
inclusion maps as
morphisms . We are given a
covariant functor
A
{\displaystyle {\mathcal {A}}}
from Mink to uC*alg , the category of
unital C* algebras, such that every morphism in Mink maps to a
monomorphism in uC*alg (isotony ).
The
Poincaré group acts
continuously on Mink . There exists a
pullback of this
action , which is continuous in the
norm topology of
A
(
M
)
{\displaystyle {\mathcal {A}}(M)}
(
Poincaré covariance ).
Minkowski space has a
causal structure . If an
open set V lies in the
causal complement of an open set U , then the
image of the maps
A
(
i
U
,
U
∪
V
)
{\displaystyle {\mathcal {A}}(i_{U,U\cup V})}
and
A
(
i
V
,
U
∪
V
)
{\displaystyle {\mathcal {A}}(i_{V,U\cup V})}
commute (spacelike commutativity). If
U
¯
{\displaystyle {\bar {U}}}
is the
causal completion of an open set U , then
A
(
i
U
,
U
¯
)
{\displaystyle {\mathcal {A}}(i_{U,{\bar {U}}})}
is an
isomorphism (primitive causality).
A
state with respect to a C*-algebra is a
positive linear functional over it with unit
norm . If we have a state over
A
(
M
)
{\displaystyle {\mathcal {A}}(M)}
, we can take the "
partial trace " to get states associated with
A
(
U
)
{\displaystyle {\mathcal {A}}(U)}
for each open set via the
net
monomorphism . The states over the open sets form a
presheaf structure.
According to the
GNS construction , for each state, we can associate a
Hilbert space
representation of
A
(
M
)
.
{\displaystyle {\mathcal {A}}(M).}
Pure states correspond to
irreducible representations and
mixed states correspond to
reducible representations . Each irreducible representation (up to
equivalence ) is called a
superselection sector . We assume there is a pure state called the
vacuum such that the Hilbert space associated with it is a
unitary representation of the
Poincaré group compatible with the Poincaré covariance of the net such that if we look at the
Poincaré algebra , the spectrum with respect to
energy-momentum (corresponding to
spacetime translations ) lies on and in the positive
light cone . This is the vacuum sector.
QFT in curved spacetime
More recently, the approach has been further implemented to include an algebraic version of
quantum field theory in curved spacetime . Indeed, the viewpoint of local quantum physics is in particular suitable to generalize the
renormalization procedure to the theory of quantum fields developed on curved backgrounds. Several rigorous results concerning QFT in presence of a
black hole have been obtained.[
citation needed ]
References
^ BaumgÀrtel, 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): 848â861,
Bibcode :
1964JMP.....5..848H ,
doi :
10.1063/1.1704187 ,
ISSN
0022-2488 ,
MR
0165864
Haag, Rudolf (1996) [1992],
Local Quantum Physics: Fields, Particles, Algebras , 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 â A New Paradigm for Local Quantum Field Theory" .
Communications in Mathematical Physics . 237 (1â2): 31â68.
arXiv :
math-ph/0112041 .
Bibcode :
2003CMaPh.237...31B .
doi :
10.1007/s00220-003-0815-7 .
S2CID
13950246 .
Brunetti, Romeo; DĂŒtsch, Michael; Fredenhagen, Klaus (2009).
"Perturbative Algebraic Quantum Field Theory and the Renormalization Groups" .
Advances in Theoretical and Mathematical Physics . 13 (5): 1541â1599.
arXiv :
0901.2038 .
doi :
10.4310/ATMP.2009.v13.n5.a7 .
S2CID
15493763 .
BĂ€r, 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 .
DĂŒtsch, Michael (2019).
From Classical Field Theory to Perturbative Quantum Field Theory . Progress in Mathematical Physics. Vol. 74. BirkhÀuser.
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
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