Algebraic constructions generated by causal structure of space-times

-algebras
*-homomorphism
Algebraic Quantum Field Theory (AQFT). Tools

Algebraic Quantum Field Theory (AQFT)
Haag-Kastler axioms
Isotony
Causality
Covariance
Time slice axiom
Spectrum condition
1,2
Haag, R., Kastler, D.: An algebraic approach to quantum field theory. J. Math. Phys. 5(7), 848–861 (1964)
1
Araki, H.: Mathematical Theory of Quantum Fields, vol. 101. Oxford UniversityPress, Oxford (1999)
2

Algebraic Quantum Field Theory (AQFT)
Isotony
Haag-Kastler axioms

Microcausality
(locality)
Algebraic Quantum Field Theory (AQFT)
Haag-Kastler axioms

Algebraic Quantum Field Theory (AQFT)
Haag-Kastler axioms

Algebraic Quantum Field Theory (AQFT)
Haag-Kastler axioms

Minkowski space-time
the family of upper cones
the family of lower cones

Operations on upper cones
addition
multiplication
x
y
z
x
y
z

Addition and multiplication on upper cones. Properties
idempotency
commutativity
associativity
absorption identity
x
y
x
y
x
=
=

Distributivity of the obtained lattice
=
x
y
z
x
y
z
=
x
y
z
x
y
z

Bijection of distributive lattices
and
Bijection T
transforms one cone into another without changing the vertex
x
y
x
y
T

Operations on diamonds
x
y
addition
multiplication
‘x
‘’x
‘y
‘’y
‘x
‘’x
‘y
‘’y

Addition and multiplication on diamonds. Properties
idempotency
commutativity
associativity
absorption identity
‘x
‘’x
‘y
‘’y
‘x
‘’x
‘y
‘’y
=
=

Distributivity of the obtained lattice
’x
‘’’x
’y
‘’’y
’’x
’’y
’x
’’’x
’y
’’’y
’’x
’’y
=
The distributive identities for the introduced operations:
addition = “union” and multiplication=“ intersection”
hold only when the involved intersections of diamonds are nonempty

Distributivity of the obtained lattice
’x
’’x
’y
’’y
’’’x
’’’y
’x
’’x
’y
’’y
’’’x
’’’y
=
The distributive identities for the introduced operations:
addition = “union” and multiplication=“intersection”
hold only when the involved intersections of diamonds are nonempty