reality construction in quantum theory: invariants and the ...qt in a nutshell & the mp...
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Reality Construction inQuantum Theory: Invariantsand the Constitutive a priori
Florian Boge
IZWT, Wuppertal
26.04.2017
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Quantum Theory in a Nutshell
I quantum theory (QT) is the empirically & practically mostsuccessful scientific theory
I underlies much of modern physics (particle physics, string theoryetc.)
I applies mostly to tiny objects (‘elementary particles’, atoms,molecules) but also to e.g. small (O(µm)) circuits (‘SQUIDS’)
I how to interpret it?
Invariants & Constitutive a priori in QT, F. Boge 2/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Quantum Theory in a Nutshell
I quantum theory (QT) is the empirically & practically mostsuccessful scientific theory
I underlies much of modern physics (particle physics, string theoryetc.)
I applies mostly to tiny objects (‘elementary particles’, atoms,molecules) but also to e.g. small (O(µm)) circuits (‘SQUIDS’)
I how to interpret it?
Invariants & Constitutive a priori in QT, F. Boge 2/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Quantum Theory in a Nutshell
I quantum theory (QT) is the empirically & practically mostsuccessful scientific theory
I underlies much of modern physics (particle physics, string theoryetc.)
I applies mostly to tiny objects (‘elementary particles’, atoms,molecules) but also to e.g. small (O(µm)) circuits (‘SQUIDS’)
I how to interpret it?
Invariants & Constitutive a priori in QT, F. Boge 2/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Quantum Theory in a Nutshell
I quantum theory (QT) is the empirically & practically mostsuccessful scientific theory
I underlies much of modern physics (particle physics, string theoryetc.)
I applies mostly to tiny objects (‘elementary particles’, atoms,molecules) but also to e.g. small (O(µm)) circuits (‘SQUIDS’)
I how to interpret it?
Invariants & Constitutive a priori in QT, F. Boge 2/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I states |ψ〉 form a linear vector space H over the field of complexnumbers, C
I C 3 α = a + ıb, where a, b ∈ R, ı =√−1
I α∗ = a − ıb is called the complex conjugateI alternate form: α = |α|eıϕ
I physically observable magnitudes (‘observables’) represented by(self-adjoint, linear) operators O acting on H
I non-commuting operators; e.g. xp − px = ı h , 0 (order makes adifference)
I unitary (linear, bijective, norm preserving) operators U representstate transformations
I e.g. dynamics: U(t; tf ) |ψ(t)〉 = |ψ(tf )〉with U(t; tf ) = e−ı h H(tf−t)
(simplest case)
Invariants & Constitutive a priori in QT, F. Boge 3/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I states |ψ〉 form a linear vector space H over the field of complexnumbers, C
I C 3 α = a + ıb, where a, b ∈ R, ı =√−1
I α∗ = a − ıb is called the complex conjugateI alternate form: α = |α|eıϕ
I physically observable magnitudes (‘observables’) represented by(self-adjoint, linear) operators O acting on H
I non-commuting operators; e.g. xp − px = ı h , 0 (order makes adifference)
I unitary (linear, bijective, norm preserving) operators U representstate transformations
I e.g. dynamics: U(t; tf ) |ψ(t)〉 = |ψ(tf )〉with U(t; tf ) = e−ı h H(tf−t)
(simplest case)
Invariants & Constitutive a priori in QT, F. Boge 3/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I states |ψ〉 form a linear vector space H over the field of complexnumbers, C
I C 3 α = a + ıb, where a, b ∈ R, ı =√−1
I α∗ = a − ıb is called the complex conjugateI alternate form: α = |α|eıϕ
I physically observable magnitudes (‘observables’) represented by(self-adjoint, linear) operators O acting on H
I non-commuting operators; e.g. xp − px = ı h , 0 (order makes adifference)
I unitary (linear, bijective, norm preserving) operators U representstate transformations
I e.g. dynamics: U(t; tf ) |ψ(t)〉 = |ψ(tf )〉with U(t; tf ) = e−ı h H(tf−t)
(simplest case)
Invariants & Constitutive a priori in QT, F. Boge 3/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I states |ψ〉 form a linear vector space H over the field of complexnumbers, C
I C 3 α = a + ıb, where a, b ∈ R, ı =√−1
I α∗ = a − ıb is called the complex conjugateI alternate form: α = |α|eıϕ
I physically observable magnitudes (‘observables’) represented by(self-adjoint, linear) operators O acting on H
I non-commuting operators; e.g. xp − px = ı h , 0 (order makes adifference)
I unitary (linear, bijective, norm preserving) operators U representstate transformations
I e.g. dynamics: U(t; tf ) |ψ(t)〉 = |ψ(tf )〉with U(t; tf ) = e−ı h H(tf−t)
(simplest case)
Invariants & Constitutive a priori in QT, F. Boge 3/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I states |ψ〉 form a linear vector space H over the field of complexnumbers, C
I C 3 α = a + ıb, where a, b ∈ R, ı =√−1
I α∗ = a − ıb is called the complex conjugateI alternate form: α = |α|eıϕ
I physically observable magnitudes (‘observables’) represented by(self-adjoint, linear) operators O acting on H
I non-commuting operators; e.g. xp − px = ı h , 0 (order makes adifference)
I unitary (linear, bijective, norm preserving) operators U representstate transformations
I e.g. dynamics: U(t; tf ) |ψ(t)〉 = |ψ(tf )〉with U(t; tf ) = e−ı h H(tf−t)
(simplest case)
Invariants & Constitutive a priori in QT, F. Boge 3/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I states |ψ〉 form a linear vector space H over the field of complexnumbers, C
I C 3 α = a + ıb, where a, b ∈ R, ı =√−1
I α∗ = a − ıb is called the complex conjugateI alternate form: α = |α|eıϕ
I physically observable magnitudes (‘observables’) represented by(self-adjoint, linear) operators O acting on H
I non-commuting operators; e.g. xp − px = ı h , 0 (order makes adifference)
I unitary (linear, bijective, norm preserving) operators U representstate transformations
I e.g. dynamics: U(t; tf ) |ψ(t)〉 = |ψ(tf )〉with U(t; tf ) = e−ı h H(tf−t)
(simplest case)
Invariants & Constitutive a priori in QT, F. Boge 3/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I states |ψ〉 form a linear vector space H over the field of complexnumbers, C
I C 3 α = a + ıb, where a, b ∈ R, ı =√−1
I α∗ = a − ıb is called the complex conjugateI alternate form: α = |α|eıϕ
I physically observable magnitudes (‘observables’) represented by(self-adjoint, linear) operators O acting on H
I non-commuting operators; e.g. xp − px = ı h , 0 (order makes adifference)
I unitary (linear, bijective, norm preserving) operators U representstate transformations
I e.g. dynamics: U(t; tf ) |ψ(t)〉 = |ψ(tf )〉with U(t; tf ) = e−ı h H(tf−t)
(simplest case)
Invariants & Constitutive a priori in QT, F. Boge 3/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I states |ψ〉 form a linear vector space H over the field of complexnumbers, C
I C 3 α = a + ıb, where a, b ∈ R, ı =√−1
I α∗ = a − ıb is called the complex conjugateI alternate form: α = |α|eıϕ
I physically observable magnitudes (‘observables’) represented by(self-adjoint, linear) operators O acting on H
I non-commuting operators; e.g. xp − px = ı h , 0 (order makes adifference)
I unitary (linear, bijective, norm preserving) operators U representstate transformations
I e.g. dynamics: U(t; tf ) |ψ(t)〉 = |ψ(tf )〉with U(t; tf ) = e−ı h H(tf−t)
(simplest case)
Invariants & Constitutive a priori in QT, F. Boge 3/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I principle of superposition (kinematic): If |a〉 , |b〉 represent states, sodoes |c〉 = α |a〉+ β |b〉 , α,β ∈ C
I ubiquitous feature: any state can be written as a ‘superpositionstate’, in a different basis of H:
Invariants & Constitutive a priori in QT, F. Boge 4/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I principle of superposition (kinematic): If |a〉 , |b〉 represent states, sodoes |c〉 = α |a〉+ β |b〉 , α,β ∈ C
I ubiquitous feature: any state can be written as a ‘superpositionstate’, in a different basis of H:
Invariants & Constitutive a priori in QT, F. Boge 4/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I principle of superposition (kinematic): If |a〉 , |b〉 represent states, sodoes |c〉 = α |a〉+ β |b〉 , α,β ∈ C
I ubiquitous feature: any state can be written as a ‘superpositionstate’, in a different basis of H:
Invariants & Constitutive a priori in QT, F. Boge 4/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Structure of the Theory
I principle of superposition (kinematic): If |a〉 , |b〉 represent states, sodoes |c〉 = α |a〉+ β |b〉 , α,β ∈ C
I ubiquitous feature: any state can be written as a ‘superpositionstate’, in a different basis of H:
|A2〉
|A1〉
|B1〉
|B2〉
Invariants & Constitutive a priori in QT, F. Boge 4/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Minimal Interpretation
I eigenvalue-eigenstate link: System S, described by |oj〉, is in a statewith definite value oj for observable O iff O |oj〉 = oj |oj〉
I non-commuting (self-adjoint) operators representnon-co-measurable observables
I for general state |ψ〉 = α1 |o1〉+ α2 |o2〉+ α3 |o3〉+ . . . =∑
j αj |oj〉,we have PrψO (oj) =
∣∣⟨oj∣∣ψ⟩∣∣2 = |αj|
2 (Born’s rule)I⟨oj∣∣ψ⟩ is called an inner product and computes a complex number
(αj)
I 〈oi|oj〉 = δij =
1 if i = j0 else
(orthonormal basis; ONB)
Invariants & Constitutive a priori in QT, F. Boge 5/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Minimal Interpretation
I eigenvalue-eigenstate link: System S, described by |oj〉, is in a statewith definite value oj for observable O iff O |oj〉 = oj |oj〉
I non-commuting (self-adjoint) operators representnon-co-measurable observables
I for general state |ψ〉 = α1 |o1〉+ α2 |o2〉+ α3 |o3〉+ . . . =∑
j αj |oj〉,we have PrψO (oj) =
∣∣⟨oj∣∣ψ⟩∣∣2 = |αj|
2 (Born’s rule)I⟨oj∣∣ψ⟩ is called an inner product and computes a complex number
(αj)
I 〈oi|oj〉 = δij =
1 if i = j0 else
(orthonormal basis; ONB)
Invariants & Constitutive a priori in QT, F. Boge 5/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Minimal Interpretation
I eigenvalue-eigenstate link: System S, described by |oj〉, is in a statewith definite value oj for observable O iff O |oj〉 = oj |oj〉
I non-commuting (self-adjoint) operators representnon-co-measurable observables
I for general state |ψ〉 = α1 |o1〉+ α2 |o2〉+ α3 |o3〉+ . . . =∑
j αj |oj〉,we have PrψO (oj) =
∣∣⟨oj∣∣ψ⟩∣∣2 = |αj|
2 (Born’s rule)I⟨oj∣∣ψ⟩ is called an inner product and computes a complex number
(αj)
I 〈oi|oj〉 = δij =
1 if i = j0 else
(orthonormal basis; ONB)
Invariants & Constitutive a priori in QT, F. Boge 5/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Minimal Interpretation
I eigenvalue-eigenstate link: System S, described by |oj〉, is in a statewith definite value oj for observable O iff O |oj〉 = oj |oj〉
I non-commuting (self-adjoint) operators representnon-co-measurable observables
I for general state |ψ〉 = α1 |o1〉+ α2 |o2〉+ α3 |o3〉+ . . . =∑
j αj |oj〉,we have PrψO (oj) =
∣∣⟨oj∣∣ψ⟩∣∣2 = |αj|
2 (Born’s rule)I⟨oj∣∣ψ⟩ is called an inner product and computes a complex number
(αj)
I 〈oi|oj〉 = δij =
1 if i = j0 else
(orthonormal basis; ONB)
Invariants & Constitutive a priori in QT, F. Boge 5/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Minimal Interpretation
I eigenvalue-eigenstate link: System S, described by |oj〉, is in a statewith definite value oj for observable O iff O |oj〉 = oj |oj〉
I non-commuting (self-adjoint) operators representnon-co-measurable observables
I for general state |ψ〉 = α1 |o1〉+ α2 |o2〉+ α3 |o3〉+ . . . =∑
j αj |oj〉,we have PrψO (oj) =
∣∣⟨oj∣∣ψ⟩∣∣2 = |αj|
2 (Born’s rule)I⟨oj∣∣ψ⟩ is called an inner product and computes a complex number
(αj)
I 〈oi|oj〉 = δij =
1 if i = j0 else
(orthonormal basis; ONB)
Invariants & Constitutive a priori in QT, F. Boge 5/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
The Measurement Problem
I |oj〉 |M0〉U7−→ |ok〉 |Moj〉, where U = e
ı h Hint∆t
I∑
j αj |oj〉 |M0〉U7−→∑
j,k αjk |ok〉 |Moj〉 =∑
j αj |oj〉 |Moj〉 =: |ψSM〉I entangled state, i.e., cannot be written as | ˜oj〉 |Moj〉 in any basis ofHS ⊗HM
I projection postulate (Dirac, 1958; von Neumann, 1932):|ψSM〉 7→ P`|ψSM〉
‖P`|ψSM〉‖= |o`〉 |Mo`〉
I what does this all mean?I how/when/where/why does the change occur? What causes it?
(“Heisenberg cut”; “Wigner’s friend”)
Invariants & Constitutive a priori in QT, F. Boge 6/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
The Measurement Problem
I |oj〉 |M0〉U7−→ |ok〉 |Moj〉, where U = e
ı h Hint∆t
I∑
j αj |oj〉 |M0〉U7−→∑
j,k αjk |ok〉 |Moj〉 =∑
j αj |oj〉 |Moj〉 =: |ψSM〉I entangled state, i.e., cannot be written as | ˜oj〉 |Moj〉 in any basis ofHS ⊗HM
I projection postulate (Dirac, 1958; von Neumann, 1932):|ψSM〉 7→ P`|ψSM〉
‖P`|ψSM〉‖= |o`〉 |Mo`〉
I what does this all mean?I how/when/where/why does the change occur? What causes it?
(“Heisenberg cut”; “Wigner’s friend”)
Invariants & Constitutive a priori in QT, F. Boge 6/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
The Measurement Problem
I |oj〉 |M0〉U7−→ |ok〉 |Moj〉, where U = e
ı h Hint∆t
I∑
j αj |oj〉 |M0〉U7−→∑
j,k αjk |ok〉 |Moj〉 =∑
j αj |oj〉 |Moj〉 =: |ψSM〉I entangled state, i.e., cannot be written as | ˜oj〉 |Moj〉 in any basis ofHS ⊗HM
I projection postulate (Dirac, 1958; von Neumann, 1932):|ψSM〉 7→ P`|ψSM〉
‖P`|ψSM〉‖= |o`〉 |Mo`〉
I what does this all mean?I how/when/where/why does the change occur? What causes it?
(“Heisenberg cut”; “Wigner’s friend”)
Invariants & Constitutive a priori in QT, F. Boge 6/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
The Measurement Problem
I |oj〉 |M0〉U7−→ |ok〉 |Moj〉, where U = e
ı h Hint∆t
I∑
j αj |oj〉 |M0〉U7−→∑
j,k αjk |ok〉 |Moj〉 =∑
j αj |oj〉 |Moj〉 =: |ψSM〉I entangled state, i.e., cannot be written as | ˜oj〉 |Moj〉 in any basis ofHS ⊗HM
I projection postulate (Dirac, 1958; von Neumann, 1932):|ψSM〉 7→ P`|ψSM〉
‖P`|ψSM〉‖= |o`〉 |Mo`〉
I what does this all mean?I how/when/where/why does the change occur? What causes it?
(“Heisenberg cut”; “Wigner’s friend”)
Invariants & Constitutive a priori in QT, F. Boge 6/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
The Measurement Problem
I |oj〉 |M0〉U7−→ |ok〉 |Moj〉, where U = e
ı h Hint∆t
I∑
j αj |oj〉 |M0〉U7−→∑
j,k αjk |ok〉 |Moj〉 =∑
j αj |oj〉 |Moj〉 =: |ψSM〉I entangled state, i.e., cannot be written as | ˜oj〉 |Moj〉 in any basis ofHS ⊗HM
I projection postulate (Dirac, 1958; von Neumann, 1932):|ψSM〉 7→ P`|ψSM〉
‖P`|ψSM〉‖= |o`〉 |Mo`〉
I what does this all mean?I how/when/where/why does the change occur? What causes it?
(“Heisenberg cut”; “Wigner’s friend”)
Invariants & Constitutive a priori in QT, F. Boge 6/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
The Measurement Problem
I |oj〉 |M0〉U7−→ |ok〉 |Moj〉, where U = e
ı h Hint∆t
I∑
j αj |oj〉 |M0〉U7−→∑
j,k αjk |ok〉 |Moj〉 =∑
j αj |oj〉 |Moj〉 =: |ψSM〉I entangled state, i.e., cannot be written as | ˜oj〉 |Moj〉 in any basis ofHS ⊗HM
I projection postulate (Dirac, 1958; von Neumann, 1932):|ψSM〉 7→ P`|ψSM〉
‖P`|ψSM〉‖= |o`〉 |Mo`〉
I what does this all mean?I how/when/where/why does the change occur? What causes it?
(“Heisenberg cut”; “Wigner’s friend”)
Invariants & Constitutive a priori in QT, F. Boge 6/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory (preliminaries)
I |ψ〉 vs. |ψ〉〈ψ| = Pψ,I |ψ〉 =
∑j |oj〉〈oj| |ψ〉 =
∑j |oj〉 〈oj|ψ〉︸ ︷︷ ︸
=αj
(pure state)
I O =∑
j oj |oj〉〈oj|
I density operator ρ =∑
k pk |ψ(k)〉〈ψ(k)| (mixed state)
Invariants & Constitutive a priori in QT, F. Boge 7/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory (preliminaries)
I |ψ〉 vs. |ψ〉〈ψ| = Pψ,I |ψ〉 =
∑j |oj〉〈oj| |ψ〉 =
∑j |oj〉 〈oj|ψ〉︸ ︷︷ ︸
=αj
(pure state)
I O =∑
j oj |oj〉〈oj|
I density operator ρ =∑
k pk |ψ(k)〉〈ψ(k)| (mixed state)
Invariants & Constitutive a priori in QT, F. Boge 7/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory (preliminaries)
I |ψ〉 vs. |ψ〉〈ψ| = Pψ,I |ψ〉 =
∑j |oj〉〈oj| |ψ〉 =
∑j |oj〉 〈oj|ψ〉︸ ︷︷ ︸
=αj
(pure state)
I O =∑
j oj |oj〉〈oj|
I density operator ρ =∑
k pk |ψ(k)〉〈ψ(k)| (mixed state)
Invariants & Constitutive a priori in QT, F. Boge 7/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory (preliminaries)
I |ψ〉 vs. |ψ〉〈ψ| = Pψ,I |ψ〉 =
∑j |oj〉〈oj| |ψ〉 =
∑j |oj〉 〈oj|ψ〉︸ ︷︷ ︸
=αj
(pure state)
I O =∑
j oj |oj〉〈oj|
I density operator ρ =∑
k pk |ψ(k)〉〈ψ(k)| (mixed state)
Invariants & Constitutive a priori in QT, F. Boge 7/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory
I USM,E |ΨSM〉 |E0〉 = USM,E∑
j αj |Sj〉 |Mj〉 |E0〉 =∑
j αj |Sj〉 |Mj〉 |Ej〉I partial tracing:
I ρSME = |ΨSME〉〈ΨSME| =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi|⊗ |Ej〉〈Ei|
I TrE(ρSME) =: ρSM =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi| 〈Ej|Ei〉
I if 〈Ei|Ej〉 ≈ 0 for i , j, after a short decoherence time we haveρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|
I approximate & improper mixture of eigenstates ( |Sj〉〈Sj|⊗ |Mj〉〈Mj|)of preferred observables M =
∑j aj |Mj〉〈Mj| , S =
∑j sj |Sj〉〈Sj|
I states stable under influence of environment (‘preferred basis’)I typically approximately localized states with approximately
well-defined velocity/momentum (quasi-classical)I not one single state of the preferred observable!
Invariants & Constitutive a priori in QT, F. Boge 8/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory
I USM,E |ΨSM〉 |E0〉 = USM,E∑
j αj |Sj〉 |Mj〉 |E0〉 =∑
j αj |Sj〉 |Mj〉 |Ej〉I partial tracing:
I ρSME = |ΨSME〉〈ΨSME| =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi|⊗ |Ej〉〈Ei|
I TrE(ρSME) =: ρSM =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi| 〈Ej|Ei〉
I if 〈Ei|Ej〉 ≈ 0 for i , j, after a short decoherence time we haveρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|
I approximate & improper mixture of eigenstates ( |Sj〉〈Sj|⊗ |Mj〉〈Mj|)of preferred observables M =
∑j aj |Mj〉〈Mj| , S =
∑j sj |Sj〉〈Sj|
I states stable under influence of environment (‘preferred basis’)I typically approximately localized states with approximately
well-defined velocity/momentum (quasi-classical)I not one single state of the preferred observable!
Invariants & Constitutive a priori in QT, F. Boge 8/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory
I USM,E |ΨSM〉 |E0〉 = USM,E∑
j αj |Sj〉 |Mj〉 |E0〉 =∑
j αj |Sj〉 |Mj〉 |Ej〉I partial tracing:
I ρSME = |ΨSME〉〈ΨSME| =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi|⊗ |Ej〉〈Ei|
I TrE(ρSME) =: ρSM =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi| 〈Ej|Ei〉
I if 〈Ei|Ej〉 ≈ 0 for i , j, after a short decoherence time we haveρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|
I approximate & improper mixture of eigenstates ( |Sj〉〈Sj|⊗ |Mj〉〈Mj|)of preferred observables M =
∑j aj |Mj〉〈Mj| , S =
∑j sj |Sj〉〈Sj|
I states stable under influence of environment (‘preferred basis’)I typically approximately localized states with approximately
well-defined velocity/momentum (quasi-classical)I not one single state of the preferred observable!
Invariants & Constitutive a priori in QT, F. Boge 8/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory
I USM,E |ΨSM〉 |E0〉 = USM,E∑
j αj |Sj〉 |Mj〉 |E0〉 =∑
j αj |Sj〉 |Mj〉 |Ej〉I partial tracing:
I ρSME = |ΨSME〉〈ΨSME| =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi|⊗ |Ej〉〈Ei|
I TrE(ρSME) =: ρSM =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi| 〈Ej|Ei〉
I if 〈Ei|Ej〉 ≈ 0 for i , j, after a short decoherence time we haveρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|
I approximate & improper mixture of eigenstates ( |Sj〉〈Sj|⊗ |Mj〉〈Mj|)of preferred observables M =
∑j aj |Mj〉〈Mj| , S =
∑j sj |Sj〉〈Sj|
I states stable under influence of environment (‘preferred basis’)I typically approximately localized states with approximately
well-defined velocity/momentum (quasi-classical)I not one single state of the preferred observable!
Invariants & Constitutive a priori in QT, F. Boge 8/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory
I USM,E |ΨSM〉 |E0〉 = USM,E∑
j αj |Sj〉 |Mj〉 |E0〉 =∑
j αj |Sj〉 |Mj〉 |Ej〉I partial tracing:
I ρSME = |ΨSME〉〈ΨSME| =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi|⊗ |Ej〉〈Ei|
I TrE(ρSME) =: ρSM =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi| 〈Ej|Ei〉
I if 〈Ei|Ej〉 ≈ 0 for i , j, after a short decoherence time we haveρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|
I approximate & improper mixture of eigenstates ( |Sj〉〈Sj|⊗ |Mj〉〈Mj|)of preferred observables M =
∑j aj |Mj〉〈Mj| , S =
∑j sj |Sj〉〈Sj|
I states stable under influence of environment (‘preferred basis’)I typically approximately localized states with approximately
well-defined velocity/momentum (quasi-classical)I not one single state of the preferred observable!
Invariants & Constitutive a priori in QT, F. Boge 8/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory
I USM,E |ΨSM〉 |E0〉 = USM,E∑
j αj |Sj〉 |Mj〉 |E0〉 =∑
j αj |Sj〉 |Mj〉 |Ej〉I partial tracing:
I ρSME = |ΨSME〉〈ΨSME| =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi|⊗ |Ej〉〈Ei|
I TrE(ρSME) =: ρSM =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi| 〈Ej|Ei〉
I if 〈Ei|Ej〉 ≈ 0 for i , j, after a short decoherence time we haveρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|
I approximate & improper mixture of eigenstates ( |Sj〉〈Sj|⊗ |Mj〉〈Mj|)of preferred observables M =
∑j aj |Mj〉〈Mj| , S =
∑j sj |Sj〉〈Sj|
I states stable under influence of environment (‘preferred basis’)I typically approximately localized states with approximately
well-defined velocity/momentum (quasi-classical)I not one single state of the preferred observable!
Invariants & Constitutive a priori in QT, F. Boge 8/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory
I USM,E |ΨSM〉 |E0〉 = USM,E∑
j αj |Sj〉 |Mj〉 |E0〉 =∑
j αj |Sj〉 |Mj〉 |Ej〉I partial tracing:
I ρSME = |ΨSME〉〈ΨSME| =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi|⊗ |Ej〉〈Ei|
I TrE(ρSME) =: ρSM =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi| 〈Ej|Ei〉
I if 〈Ei|Ej〉 ≈ 0 for i , j, after a short decoherence time we haveρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|
I approximate & improper mixture of eigenstates ( |Sj〉〈Sj|⊗ |Mj〉〈Mj|)of preferred observables M =
∑j aj |Mj〉〈Mj| , S =
∑j sj |Sj〉〈Sj|
I states stable under influence of environment (‘preferred basis’)I typically approximately localized states with approximately
well-defined velocity/momentum (quasi-classical)I not one single state of the preferred observable!
Invariants & Constitutive a priori in QT, F. Boge 8/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory
I USM,E |ΨSM〉 |E0〉 = USM,E∑
j αj |Sj〉 |Mj〉 |E0〉 =∑
j αj |Sj〉 |Mj〉 |Ej〉I partial tracing:
I ρSME = |ΨSME〉〈ΨSME| =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi|⊗ |Ej〉〈Ei|
I TrE(ρSME) =: ρSM =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi| 〈Ej|Ei〉
I if 〈Ei|Ej〉 ≈ 0 for i , j, after a short decoherence time we haveρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|
I approximate & improper mixture of eigenstates ( |Sj〉〈Sj|⊗ |Mj〉〈Mj|)of preferred observables M =
∑j aj |Mj〉〈Mj| , S =
∑j sj |Sj〉〈Sj|
I states stable under influence of environment (‘preferred basis’)I typically approximately localized states with approximately
well-defined velocity/momentum (quasi-classical)I not one single state of the preferred observable!
Invariants & Constitutive a priori in QT, F. Boge 8/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Decoherence Theory
I USM,E |ΨSM〉 |E0〉 = USM,E∑
j αj |Sj〉 |Mj〉 |E0〉 =∑
j αj |Sj〉 |Mj〉 |Ej〉I partial tracing:
I ρSME = |ΨSME〉〈ΨSME| =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi|⊗ |Ej〉〈Ei|
I TrE(ρSME) =: ρSM =∑
i,j αjα∗i |Sj〉〈Si|⊗ |Mj〉〈Mi| 〈Ej|Ei〉
I if 〈Ei|Ej〉 ≈ 0 for i , j, after a short decoherence time we haveρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|
I approximate & improper mixture of eigenstates ( |Sj〉〈Sj|⊗ |Mj〉〈Mj|)of preferred observables M =
∑j aj |Mj〉〈Mj| , S =
∑j sj |Sj〉〈Sj|
I states stable under influence of environment (‘preferred basis’)I typically approximately localized states with approximately
well-defined velocity/momentum (quasi-classical)I not one single state of the preferred observable!
Invariants & Constitutive a priori in QT, F. Boge 8/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I time evolution unitary, does not change the norm:〈ψ|U†(t)U(t)|ψ〉 = 〈ψ|ψ〉
I with |ψ(t)〉 =∑
j αj(t) |oj〉 this means∑i,j α∗i (t)αj(t) 〈oi|oj〉︸ ︷︷ ︸
=δij
=∑
j |αj(t)|2
I probability conservation
Invariants & Constitutive a priori in QT, F. Boge 9/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I time evolution unitary, does not change the norm:〈ψ|U†(t)U(t)|ψ〉 = 〈ψ|ψ〉
I with |ψ(t)〉 =∑
j αj(t) |oj〉 this means∑i,j α∗i (t)αj(t) 〈oi|oj〉︸ ︷︷ ︸
=δij
=∑
j |αj(t)|2
I probability conservation
Invariants & Constitutive a priori in QT, F. Boge 9/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I time evolution unitary, does not change the norm:〈ψ|U†(t)U(t)|ψ〉 = 〈ψ|ψ〉
I with |ψ(t)〉 =∑
j αj(t) |oj〉 this means∑i,j α∗i (t)αj(t) 〈oi|oj〉︸ ︷︷ ︸
=δij
=∑
j |αj(t)|2
I probability conservation
Invariants & Constitutive a priori in QT, F. Boge 9/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I objective probabilities?
I projection postulate instantaneous action at a distanceI what about relativity; preferred frame of reference?
I epistemic probabilities?
Invariants & Constitutive a priori in QT, F. Boge 10/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I objective probabilities?
I projection postulate instantaneous action at a distanceI what about relativity; preferred frame of reference?
I epistemic probabilities?
Invariants & Constitutive a priori in QT, F. Boge 10/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I objective probabilities?
|X1〉 |X2〉 |X3〉 |X4〉Pi |X3〉
I projection postulate instantaneous action at a distanceI what about relativity; preferred frame of reference?
I epistemic probabilities?
Invariants & Constitutive a priori in QT, F. Boge 10/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I objective probabilities?
|X1〉 |X2〉 |X3〉 |X4〉Pi |X3〉
I projection postulate instantaneous action at a distanceI what about relativity; preferred frame of reference?
I epistemic probabilities?
Invariants & Constitutive a priori in QT, F. Boge 10/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I objective probabilities?
|X1〉 |X2〉 |X3〉 |X4〉Pi |X3〉
I projection postulate instantaneous action at a distanceI what about relativity; preferred frame of reference?
I epistemic probabilities?
Invariants & Constitutive a priori in QT, F. Boge 10/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I objective probabilities?
|X1〉 |X2〉 |X3〉 |X4〉Pi |X3〉
I projection postulate instantaneous action at a distanceI what about relativity; preferred frame of reference?
I epistemic probabilities?
Invariants & Constitutive a priori in QT, F. Boge 10/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I objective probabilities?
|X1〉 |X2〉 |X3〉 |X4〉Pi |X3〉
I projection postulate instantaneous action at a distanceI what about relativity; preferred frame of reference?
t = 0
t t′
t′ = 0
causescausesprevents
I epistemic probabilities?Invariants & Constitutive a priori in QT, F. Boge 10/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Meaning of the Probabilities?
I objective probabilities?
|X1〉 |X2〉 |X3〉 |X4〉Pi |X3〉
I projection postulate instantaneous action at a distanceI what about relativity; preferred frame of reference?
t = 0
t t′
t′ = 0
causescausesprevents
I epistemic probabilities?Invariants & Constitutive a priori in QT, F. Boge 10/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Hidden Variables?
I assumption: hidden states λ that do not occur in QT (Einstein et al.,1935)
I modern framework (Spekkens, 2005; Harrigan and Spekkens,2010): prepared state |ψ〉 7→ pψ(λ), measured state |φ〉 7→ ξ
φM(λ)
I | 〈φ|ψ〉 |2 =∫Λ dλ pψ(λ)ξ
φM(λ)
I can so far only reproduce parts of QT (e.g. Spekkens, 2007; Bartlettet al., 2012)
I incompatible with a host of other desirable assumptions about λ(e.g. Pusey et al., 2012; Hardy, 2013)
Invariants & Constitutive a priori in QT, F. Boge 11/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Hidden Variables?
I assumption: hidden states λ that do not occur in QT (Einstein et al.,1935)
I modern framework (Spekkens, 2005; Harrigan and Spekkens,2010): prepared state |ψ〉 7→ pψ(λ), measured state |φ〉 7→ ξ
φM(λ)
I | 〈φ|ψ〉 |2 =∫Λ dλ pψ(λ)ξ
φM(λ)
I can so far only reproduce parts of QT (e.g. Spekkens, 2007; Bartlettet al., 2012)
I incompatible with a host of other desirable assumptions about λ(e.g. Pusey et al., 2012; Hardy, 2013)
Invariants & Constitutive a priori in QT, F. Boge 11/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Hidden Variables?
I assumption: hidden states λ that do not occur in QT (Einstein et al.,1935)
I modern framework (Spekkens, 2005; Harrigan and Spekkens,2010): prepared state |ψ〉 7→ pψ(λ), measured state |φ〉 7→ ξ
φM(λ)
I | 〈φ|ψ〉 |2 =∫Λ dλ pψ(λ)ξ
φM(λ)
I can so far only reproduce parts of QT (e.g. Spekkens, 2007; Bartlettet al., 2012)
I incompatible with a host of other desirable assumptions about λ(e.g. Pusey et al., 2012; Hardy, 2013)
Invariants & Constitutive a priori in QT, F. Boge 11/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Hidden Variables?
I assumption: hidden states λ that do not occur in QT (Einstein et al.,1935)
I modern framework (Spekkens, 2005; Harrigan and Spekkens,2010): prepared state |ψ〉 7→ pψ(λ), measured state |φ〉 7→ ξ
φM(λ)
I | 〈φ|ψ〉 |2 =∫Λ dλ pψ(λ)ξ
φM(λ)
I can so far only reproduce parts of QT (e.g. Spekkens, 2007; Bartlettet al., 2012)
I incompatible with a host of other desirable assumptions about λ(e.g. Pusey et al., 2012; Hardy, 2013)
Invariants & Constitutive a priori in QT, F. Boge 11/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Hidden Variables?
I assumption: hidden states λ that do not occur in QT (Einstein et al.,1935)
I modern framework (Spekkens, 2005; Harrigan and Spekkens,2010): prepared state |ψ〉 7→ pψ(λ), measured state |φ〉 7→ ξ
φM(λ)
I | 〈φ|ψ〉 |2 =∫Λ dλ pψ(λ)ξ
φM(λ)
I can so far only reproduce parts of QT (e.g. Spekkens, 2007; Bartlettet al., 2012)
I incompatible with a host of other desirable assumptions about λ(e.g. Pusey et al., 2012; Hardy, 2013)
Invariants & Constitutive a priori in QT, F. Boge 11/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
I most important result: Bell inequalities (Bell, 1964)a
b
θab
I Prψ(Lxi , Ry
j |i, j, λ) = Prψ(Lxi |i, λ)Prψ(R
yj |j, λ)
I Prψ(λ|i, j) = Prψ(λ)⇒ −2 6
⟨Rx
j , Lyi
⟩i,j,ψ
+⟨
Rxj , Ly
i′
⟩i′,j,ψ
+⟨
Rxj′ , Ly
i
⟩i,j′,ψ
−⟨
Rxj′ , Ly
i′
⟩i′,j′,ψ
6 2
I violated according to |χ〉 = 1√2(|↑↓〉− |↓↑〉) – which implies
probabilities 12 cos2(θab/2) for respective opposite values and
12 sin2(θab/2) for respective equal values – and experiment
Invariants & Constitutive a priori in QT, F. Boge 12/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
I most important result: Bell inequalities (Bell, 1964)a
b
θab
I Prψ(Lxi , Ry
j |i, j, λ) = Prψ(Lxi |i, λ)Prψ(R
yj |j, λ)
I Prψ(λ|i, j) = Prψ(λ)⇒ −2 6
⟨Rx
j , Lyi
⟩i,j,ψ
+⟨
Rxj , Ly
i′
⟩i′,j,ψ
+⟨
Rxj′ , Ly
i
⟩i,j′,ψ
−⟨
Rxj′ , Ly
i′
⟩i′,j′,ψ
6 2
I violated according to |χ〉 = 1√2(|↑↓〉− |↓↑〉) – which implies
probabilities 12 cos2(θab/2) for respective opposite values and
12 sin2(θab/2) for respective equal values – and experiment
Invariants & Constitutive a priori in QT, F. Boge 12/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
I most important result: Bell inequalities (Bell, 1964)a
b
θab
I Prψ(Lxi , Ry
j |i, j, λ) = Prψ(Lxi |i, λ)Prψ(R
yj |j, λ)
I Prψ(λ|i, j) = Prψ(λ)⇒ −2 6
⟨Rx
j , Lyi
⟩i,j,ψ
+⟨
Rxj , Ly
i′
⟩i′,j,ψ
+⟨
Rxj′ , Ly
i
⟩i,j′,ψ
−⟨
Rxj′ , Ly
i′
⟩i′,j′,ψ
6 2
I violated according to |χ〉 = 1√2(|↑↓〉− |↓↑〉) – which implies
probabilities 12 cos2(θab/2) for respective opposite values and
12 sin2(θab/2) for respective equal values – and experiment
Invariants & Constitutive a priori in QT, F. Boge 12/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
I most important result: Bell inequalities (Bell, 1964)a
b
θab
I Prψ(Lxi , Ry
j |i, j, λ) = Prψ(Lxi |i, λ)Prψ(R
yj |j, λ)
I Prψ(λ|i, j) = Prψ(λ)⇒ −2 6
⟨Rx
j , Lyi
⟩i,j,ψ
+⟨
Rxj , Ly
i′
⟩i′,j,ψ
+⟨
Rxj′ , Ly
i
⟩i,j′,ψ
−⟨
Rxj′ , Ly
i′
⟩i′,j′,ψ
6 2
I violated according to |χ〉 = 1√2(|↑↓〉− |↓↑〉) – which implies
probabilities 12 cos2(θab/2) for respective opposite values and
12 sin2(θab/2) for respective equal values – and experiment
Invariants & Constitutive a priori in QT, F. Boge 12/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
I most important result: Bell inequalities (Bell, 1964)a
b
θab
I Prψ(Lxi , Ry
j |i, j, λ) = Prψ(Lxi |i, λ)Prψ(R
yj |j, λ)
I Prψ(λ|i, j) = Prψ(λ)⇒ −2 6
⟨Rx
j , Lyi
⟩i,j,ψ
+⟨
Rxj , Ly
i′
⟩i′,j,ψ
+⟨
Rxj′ , Ly
i
⟩i,j′,ψ
−⟨
Rxj′ , Ly
i′
⟩i′,j′,ψ
6 2
I violated according to |χ〉 = 1√2(|↑↓〉− |↓↑〉) – which implies
probabilities 12 cos2(θab/2) for respective opposite values and
12 sin2(θab/2) for respective equal values – and experiment
Invariants & Constitutive a priori in QT, F. Boge 12/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
I most important result: Bell inequalities (Bell, 1964)a
b
θab
I Prψ(Lxi , Ry
j |i, j, λ) = Prψ(Lxi |i, λ)Prψ(R
yj |j, λ)
I Prψ(λ|i, j) = Prψ(λ)⇒ −2 6
⟨Rx
j , Lyi
⟩i,j,ψ
+⟨
Rxj , Ly
i′
⟩i′,j,ψ
+⟨
Rxj′ , Ly
i
⟩i,j′,ψ
−⟨
Rxj′ , Ly
i′
⟩i′,j′,ψ
6 2
I violated according to |χ〉 = 1√2(|↑↓〉− |↓↑〉) – which implies
probabilities 12 cos2(θab/2) for respective opposite values and
12 sin2(θab/2) for respective equal values – and experiment
Invariants & Constitutive a priori in QT, F. Boge 12/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
D1 D2
s1 s2
E
t
x
λ
Invariants & Constitutive a priori in QT, F. Boge 13/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
I Einstein (1949): “either [...] the measurement of S1 (telepathically)changes the real situation of S2 or [one has to deny] independentreal situations as such to things which are spatially separated fromeach other. Both alternatives appear to me entirely unacceptable.”
I Bohr (1935) demands “a final renunciation of the classical ideal ofcausality and a radical revision of our attitude towards theproblem of physical reality.”
Invariants & Constitutive a priori in QT, F. Boge 14/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
I Einstein (1949): “either [...] the measurement of S1 (telepathically)changes the real situation of S2 or [one has to deny] independentreal situations as such to things which are spatially separated fromeach other. Both alternatives appear to me entirely unacceptable.”
I Bohr (1935) demands “a final renunciation of the classical ideal ofcausality and a radical revision of our attitude towards theproblem of physical reality.”
Invariants & Constitutive a priori in QT, F. Boge 14/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Copenhagen traditionI ambiguous/hard to pin down as a precise interpretation at all
I QBismI offers theorems that connect probabilities in QT to subjective
Bayesian onesI relies on rather objective assumptions after all (‘frequency data’;
implicit rationality constraints for agents)I cannot make sense of observed frequenciesI does not account well for decoherenceI vague ond epistemological issues
Invariants & Constitutive a priori in QT, F. Boge 15/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Copenhagen traditionI ambiguous/hard to pin down as a precise interpretation at all
I QBismI offers theorems that connect probabilities in QT to subjective
Bayesian onesI relies on rather objective assumptions after all (‘frequency data’;
implicit rationality constraints for agents)I cannot make sense of observed frequenciesI does not account well for decoherenceI vague ond epistemological issues
Invariants & Constitutive a priori in QT, F. Boge 15/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Copenhagen traditionI ambiguous/hard to pin down as a precise interpretation at all
I QBismI offers theorems that connect probabilities in QT to subjective
Bayesian onesI relies on rather objective assumptions after all (‘frequency data’;
implicit rationality constraints for agents)I cannot make sense of observed frequenciesI does not account well for decoherenceI vague ond epistemological issues
Invariants & Constitutive a priori in QT, F. Boge 15/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Copenhagen traditionI ambiguous/hard to pin down as a precise interpretation at all
I QBismI offers theorems that connect probabilities in QT to subjective
Bayesian onesI relies on rather objective assumptions after all (‘frequency data’;
implicit rationality constraints for agents)I cannot make sense of observed frequenciesI does not account well for decoherenceI vague ond epistemological issues
Invariants & Constitutive a priori in QT, F. Boge 15/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Copenhagen traditionI ambiguous/hard to pin down as a precise interpretation at all
I QBismI offers theorems that connect probabilities in QT to subjective
Bayesian onesI relies on rather objective assumptions after all (‘frequency data’;
implicit rationality constraints for agents)I cannot make sense of observed frequenciesI does not account well for decoherenceI vague ond epistemological issues
Invariants & Constitutive a priori in QT, F. Boge 15/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Copenhagen traditionI ambiguous/hard to pin down as a precise interpretation at all
I QBismI offers theorems that connect probabilities in QT to subjective
Bayesian onesI relies on rather objective assumptions after all (‘frequency data’;
implicit rationality constraints for agents)I cannot make sense of observed frequenciesI does not account well for decoherenceI vague ond epistemological issues
Invariants & Constitutive a priori in QT, F. Boge 15/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Copenhagen traditionI ambiguous/hard to pin down as a precise interpretation at all
I QBismI offers theorems that connect probabilities in QT to subjective
Bayesian onesI relies on rather objective assumptions after all (‘frequency data’;
implicit rationality constraints for agents)I cannot make sense of observed frequenciesI does not account well for decoherenceI vague ond epistemological issues
Invariants & Constitutive a priori in QT, F. Boge 15/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Copenhagen traditionI ambiguous/hard to pin down as a precise interpretation at all
I QBismI offers theorems that connect probabilities in QT to subjective
Bayesian onesI relies on rather objective assumptions after all (‘frequency data’;
implicit rationality constraints for agents)I cannot make sense of observed frequenciesI does not account well for decoherenceI vague ond epistemological issues
Invariants & Constitutive a priori in QT, F. Boge 15/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Healey (2012)I |ψ〉 has a normative character (what should you expect), not a
descriptive oneI decoherence tells us...
(i) ...when to expect what and with what probability(ρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|)(ii) ...when certain statements are meaningful
I (ii) is too strong: we still understand “both atoms already had theirspin up”, even if not ‘warranted’ by evidence
Invariants & Constitutive a priori in QT, F. Boge 16/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Healey (2012)I |ψ〉 has a normative character (what should you expect), not a
descriptive oneI decoherence tells us...
(i) ...when to expect what and with what probability(ρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|)(ii) ...when certain statements are meaningful
I (ii) is too strong: we still understand “both atoms already had theirspin up”, even if not ‘warranted’ by evidence
Invariants & Constitutive a priori in QT, F. Boge 16/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Healey (2012)I |ψ〉 has a normative character (what should you expect), not a
descriptive oneI decoherence tells us...
(i) ...when to expect what and with what probability(ρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|)(ii) ...when certain statements are meaningful
I (ii) is too strong: we still understand “both atoms already had theirspin up”, even if not ‘warranted’ by evidence
Invariants & Constitutive a priori in QT, F. Boge 16/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Healey (2012)I |ψ〉 has a normative character (what should you expect), not a
descriptive oneI decoherence tells us...
(i) ...when to expect what and with what probability(ρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|)(ii) ...when certain statements are meaningful
I (ii) is too strong: we still understand “both atoms already had theirspin up”, even if not ‘warranted’ by evidence
Invariants & Constitutive a priori in QT, F. Boge 16/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Healey (2012)I |ψ〉 has a normative character (what should you expect), not a
descriptive oneI decoherence tells us...
(i) ...when to expect what and with what probability(ρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|)(ii) ...when certain statements are meaningful
I (ii) is too strong: we still understand “both atoms already had theirspin up”, even if not ‘warranted’ by evidence
Invariants & Constitutive a priori in QT, F. Boge 16/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Epistemic Interpretations Without λ?
I Healey (2012)I |ψ〉 has a normative character (what should you expect), not a
descriptive oneI decoherence tells us...
(i) ...when to expect what and with what probability(ρSM ≈
∑j |αj|
2 |Sj〉〈Sj|⊗ |Mj〉〈Mj|)(ii) ...when certain statements are meaningful
I (ii) is too strong: we still understand “both atoms already had theirspin up”, even if not ‘warranted’ by evidence
Invariants & Constitutive a priori in QT, F. Boge 16/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Quine (1939, pp. 702-3; emphasis mine)If the word “Pegasus” designates something then there is such a thingas Pegasus, whereas if the word does not designate anything then thestatement would appear to lack subject-matter and thus to fall intomeaninglessness. Actually, this problem rests only on failure toobserve that a noun can be meaningful in the absence of a designatum.[...] The understanding of a term [...] does not imply a designatum; itprecedes knowledge of whether or not the term has a designatum.
Healey (2012, p. 747; emphasis mine)[...] no natural limit such that one could say that [...] one has finallysucceeded in establishing a kind of natural language-worldcorrespondence relation in virtue of which [a given] statement correctlyrepresents some radically mind- and language-independent state ofaffairs.
Invariants & Constitutive a priori in QT, F. Boge 17/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Quine (1939, pp. 702-3; emphasis mine)If the word “Pegasus” designates something then there is such a thingas Pegasus, whereas if the word does not designate anything then thestatement would appear to lack subject-matter and thus to fall intomeaninglessness. Actually, this problem rests only on failure toobserve that a noun can be meaningful in the absence of a designatum.[...] The understanding of a term [...] does not imply a designatum; itprecedes knowledge of whether or not the term has a designatum.
Healey (2012, p. 747; emphasis mine)[...] no natural limit such that one could say that [...] one has finallysucceeded in establishing a kind of natural language-worldcorrespondence relation in virtue of which [a given] statement correctlyrepresents some radically mind- and language-independent state ofaffairs.
Invariants & Constitutive a priori in QT, F. Boge 17/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I if there is no “natural language-world correspondence”, thenwhat is meant by “physical reality” (as used by Healey)?
I internal realism(s) (Kant, 1781; Putnam, 1975, 1990):I the world we experience is, to a large extent, constructed with the
aid of a priori prescriptions
I Kant’s problem: synthetic a priori with eternal, unrestricted validityis hardly tenable (e.g. Reichenbach, 1920; d’Espagnat, 2011)
I Reichenbach (1920); Friedman (1999): constitutive a priori:I relative to a given theory, there are ‘coordinative axioms’ that
prescribe what the theory talks aboutI only in virtue of these do (the terms appearing in) other statements
of the theory (‘empirical laws’) get their very meaning
Invariants & Constitutive a priori in QT, F. Boge 18/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I if there is no “natural language-world correspondence”, thenwhat is meant by “physical reality” (as used by Healey)?
I internal realism(s) (Kant, 1781; Putnam, 1975, 1990):I the world we experience is, to a large extent, constructed with the
aid of a priori prescriptions
I Kant’s problem: synthetic a priori with eternal, unrestricted validityis hardly tenable (e.g. Reichenbach, 1920; d’Espagnat, 2011)
I Reichenbach (1920); Friedman (1999): constitutive a priori:I relative to a given theory, there are ‘coordinative axioms’ that
prescribe what the theory talks aboutI only in virtue of these do (the terms appearing in) other statements
of the theory (‘empirical laws’) get their very meaning
Invariants & Constitutive a priori in QT, F. Boge 18/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I if there is no “natural language-world correspondence”, thenwhat is meant by “physical reality” (as used by Healey)?
I internal realism(s) (Kant, 1781; Putnam, 1975, 1990):I the world we experience is, to a large extent, constructed with the
aid of a priori prescriptions
I Kant’s problem: synthetic a priori with eternal, unrestricted validityis hardly tenable (e.g. Reichenbach, 1920; d’Espagnat, 2011)
I Reichenbach (1920); Friedman (1999): constitutive a priori:I relative to a given theory, there are ‘coordinative axioms’ that
prescribe what the theory talks aboutI only in virtue of these do (the terms appearing in) other statements
of the theory (‘empirical laws’) get their very meaning
Invariants & Constitutive a priori in QT, F. Boge 18/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I if there is no “natural language-world correspondence”, thenwhat is meant by “physical reality” (as used by Healey)?
I internal realism(s) (Kant, 1781; Putnam, 1975, 1990):I the world we experience is, to a large extent, constructed with the
aid of a priori prescriptions
I Kant’s problem: synthetic a priori with eternal, unrestricted validityis hardly tenable (e.g. Reichenbach, 1920; d’Espagnat, 2011)
I Reichenbach (1920); Friedman (1999): constitutive a priori:I relative to a given theory, there are ‘coordinative axioms’ that
prescribe what the theory talks aboutI only in virtue of these do (the terms appearing in) other statements
of the theory (‘empirical laws’) get their very meaning
Invariants & Constitutive a priori in QT, F. Boge 18/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I if there is no “natural language-world correspondence”, thenwhat is meant by “physical reality” (as used by Healey)?
I internal realism(s) (Kant, 1781; Putnam, 1975, 1990):I the world we experience is, to a large extent, constructed with the
aid of a priori prescriptions
I Kant’s problem: synthetic a priori with eternal, unrestricted validityis hardly tenable (e.g. Reichenbach, 1920; d’Espagnat, 2011)
I Reichenbach (1920); Friedman (1999): constitutive a priori:I relative to a given theory, there are ‘coordinative axioms’ that
prescribe what the theory talks aboutI only in virtue of these do (the terms appearing in) other statements
of the theory (‘empirical laws’) get their very meaning
Invariants & Constitutive a priori in QT, F. Boge 18/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I if there is no “natural language-world correspondence”, thenwhat is meant by “physical reality” (as used by Healey)?
I internal realism(s) (Kant, 1781; Putnam, 1975, 1990):I the world we experience is, to a large extent, constructed with the
aid of a priori prescriptions
I Kant’s problem: synthetic a priori with eternal, unrestricted validityis hardly tenable (e.g. Reichenbach, 1920; d’Espagnat, 2011)
I Reichenbach (1920); Friedman (1999): constitutive a priori:I relative to a given theory, there are ‘coordinative axioms’ that
prescribe what the theory talks aboutI only in virtue of these do (the terms appearing in) other statements
of the theory (‘empirical laws’) get their very meaning
Invariants & Constitutive a priori in QT, F. Boge 18/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I if there is no “natural language-world correspondence”, thenwhat is meant by “physical reality” (as used by Healey)?
I internal realism(s) (Kant, 1781; Putnam, 1975, 1990):I the world we experience is, to a large extent, constructed with the
aid of a priori prescriptions
I Kant’s problem: synthetic a priori with eternal, unrestricted validityis hardly tenable (e.g. Reichenbach, 1920; d’Espagnat, 2011)
I Reichenbach (1920); Friedman (1999): constitutive a priori:I relative to a given theory, there are ‘coordinative axioms’ that
prescribe what the theory talks aboutI only in virtue of these do (the terms appearing in) other statements
of the theory (‘empirical laws’) get their very meaning
Invariants & Constitutive a priori in QT, F. Boge 18/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I how to tell what is being prescribed? Find invariants of the theoryunder a relevant group, G, of transformations (e.g. Friedman, 1999,p. 67)
I example:I in general relativity, spacetime is represented by an ordered tuple(M, T(i))i∈I, where M can be covered by subsets U ⊆M that can bemapped continuously into Rn
I if two ordered tuples (M, T(i))i∈I, (N, T(i))i∈I are related by adiffeomorphism φ, a C∞ bijective map, with induced map φ∗ thatrelates T(i) = φ∗T(i), they represent the same spacetime(diffeomorphism invariance)
I “[...] and so only the underlying topology and manifold structureremain constitutively a priori. [...] physical geometry (the metric ofphysical space) is no longer constitutive.” (Friedman, 1999, p. 66)
Invariants & Constitutive a priori in QT, F. Boge 19/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I how to tell what is being prescribed? Find invariants of the theoryunder a relevant group, G, of transformations (e.g. Friedman, 1999,p. 67)
I example:I in general relativity, spacetime is represented by an ordered tuple(M, T(i))i∈I, where M can be covered by subsets U ⊆M that can bemapped continuously into Rn
I if two ordered tuples (M, T(i))i∈I, (N, T(i))i∈I are related by adiffeomorphism φ, a C∞ bijective map, with induced map φ∗ thatrelates T(i) = φ∗T(i), they represent the same spacetime(diffeomorphism invariance)
I “[...] and so only the underlying topology and manifold structureremain constitutively a priori. [...] physical geometry (the metric ofphysical space) is no longer constitutive.” (Friedman, 1999, p. 66)
Invariants & Constitutive a priori in QT, F. Boge 19/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I how to tell what is being prescribed? Find invariants of the theoryunder a relevant group, G, of transformations (e.g. Friedman, 1999,p. 67)
I example:I in general relativity, spacetime is represented by an ordered tuple(M, T(i))i∈I, where M can be covered by subsets U ⊆M that can bemapped continuously into Rn
I if two ordered tuples (M, T(i))i∈I, (N, T(i))i∈I are related by adiffeomorphism φ, a C∞ bijective map, with induced map φ∗ thatrelates T(i) = φ∗T(i), they represent the same spacetime(diffeomorphism invariance)
I “[...] and so only the underlying topology and manifold structureremain constitutively a priori. [...] physical geometry (the metric ofphysical space) is no longer constitutive.” (Friedman, 1999, p. 66)
Invariants & Constitutive a priori in QT, F. Boge 19/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I how to tell what is being prescribed? Find invariants of the theoryunder a relevant group, G, of transformations (e.g. Friedman, 1999,p. 67)
I example:I in general relativity, spacetime is represented by an ordered tuple(M, T(i))i∈I, where M can be covered by subsets U ⊆M that can bemapped continuously into Rn
I if two ordered tuples (M, T(i))i∈I, (N, T(i))i∈I are related by adiffeomorphism φ, a C∞ bijective map, with induced map φ∗ thatrelates T(i) = φ∗T(i), they represent the same spacetime(diffeomorphism invariance)
I “[...] and so only the underlying topology and manifold structureremain constitutively a priori. [...] physical geometry (the metric ofphysical space) is no longer constitutive.” (Friedman, 1999, p. 66)
Invariants & Constitutive a priori in QT, F. Boge 19/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Physical Reality
I how to tell what is being prescribed? Find invariants of the theoryunder a relevant group, G, of transformations (e.g. Friedman, 1999,p. 67)
I example:I in general relativity, spacetime is represented by an ordered tuple(M, T(i))i∈I, where M can be covered by subsets U ⊆M that can bemapped continuously into Rn
I if two ordered tuples (M, T(i))i∈I, (N, T(i))i∈I are related by adiffeomorphism φ, a C∞ bijective map, with induced map φ∗ thatrelates T(i) = φ∗T(i), they represent the same spacetime(diffeomorphism invariance)
I “[...] and so only the underlying topology and manifold structureremain constitutively a priori. [...] physical geometry (the metric ofphysical space) is no longer constitutive.” (Friedman, 1999, p. 66)
Invariants & Constitutive a priori in QT, F. Boge 19/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objects (Mittelstaedt, 1995, 2009):I in non-relativistic QT: G10 7→
Uj
j∈JI projectors Pj that pertain to a system S describe its propertiesI but: non-commutativity no well-localized objects with
simultaneously well-defined momentaI object as the carrier of a ‘non-distributive lattice’ of Pi, in all
situations that can be constructed via the UjI generalization to (special) relativistic QT: Poincare group instead of
G10 (e.g. Streater, 1988, p. l44)
Invariants & Constitutive a priori in QT, F. Boge 20/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objects (Mittelstaedt, 1995, 2009):I in non-relativistic QT: G10 7→
Uj
j∈JI projectors Pj that pertain to a system S describe its propertiesI but: non-commutativity no well-localized objects with
simultaneously well-defined momentaI object as the carrier of a ‘non-distributive lattice’ of Pi, in all
situations that can be constructed via the UjI generalization to (special) relativistic QT: Poincare group instead of
G10 (e.g. Streater, 1988, p. l44)
Invariants & Constitutive a priori in QT, F. Boge 20/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objects (Mittelstaedt, 1995, 2009):I in non-relativistic QT: G10 7→
Uj
j∈JI projectors Pj that pertain to a system S describe its propertiesI but: non-commutativity no well-localized objects with
simultaneously well-defined momentaI object as the carrier of a ‘non-distributive lattice’ of Pi, in all
situations that can be constructed via the UjI generalization to (special) relativistic QT: Poincare group instead of
G10 (e.g. Streater, 1988, p. l44)
Invariants & Constitutive a priori in QT, F. Boge 20/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objects (Mittelstaedt, 1995, 2009):I in non-relativistic QT: G10 7→
Uj
j∈JI projectors Pj that pertain to a system S describe its propertiesI but: non-commutativity no well-localized objects with
simultaneously well-defined momentaI object as the carrier of a ‘non-distributive lattice’ of Pi, in all
situations that can be constructed via the UjI generalization to (special) relativistic QT: Poincare group instead of
G10 (e.g. Streater, 1988, p. l44)
Invariants & Constitutive a priori in QT, F. Boge 20/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objects (Mittelstaedt, 1995, 2009):I in non-relativistic QT: G10 7→
Uj
j∈JI projectors Pj that pertain to a system S describe its propertiesI but: non-commutativity no well-localized objects with
simultaneously well-defined momentaI object as the carrier of a ‘non-distributive lattice’ of Pi, in all
situations that can be constructed via the UjI generalization to (special) relativistic QT: Poincare group instead of
G10 (e.g. Streater, 1988, p. l44)
Invariants & Constitutive a priori in QT, F. Boge 20/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objects (Mittelstaedt, 1995, 2009):I in non-relativistic QT: G10 7→
Uj
j∈JI projectors Pj that pertain to a system S describe its propertiesI but: non-commutativity no well-localized objects with
simultaneously well-defined momentaI object as the carrier of a ‘non-distributive lattice’ of Pi, in all
situations that can be constructed via the UjI generalization to (special) relativistic QT: Poincare group instead of
G10 (e.g. Streater, 1988, p. l44)
Invariants & Constitutive a priori in QT, F. Boge 20/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I point particles (Falkenburg, 2007, S. 134 ff.):I (differential) cross section, e.g.
dσdΩ ∝
cos2(θ2 )
sin6(θ2 )
[F2
1 −q2
4M2 [2(F1 + 2MF2)2 tan2(θ2 ) + (2MF2)
2]]
(Rosenbluth; cf. Drell and Zachariasen 1961, p. 8)I Fi = Fi(q2) come from | 〈pout|T|pin〉 |2I “a particle has [...] structure – i.e. is not a point particle – if and only
if the functions F1(q2) and/or F2(q2) are not constant.” (Drell andZachariasen, 1961, ibid.; emphasis mine)
Invariants & Constitutive a priori in QT, F. Boge 21/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I point particles (Falkenburg, 2007, S. 134 ff.):I (differential) cross section, e.g.
dσdΩ ∝
cos2(θ2 )
sin6(θ2 )
[F2
1 −q2
4M2 [2(F1 + 2MF2)2 tan2(θ2 ) + (2MF2)
2]]
(Rosenbluth; cf. Drell and Zachariasen 1961, p. 8)I Fi = Fi(q2) come from | 〈pout|T|pin〉 |2I “a particle has [...] structure – i.e. is not a point particle – if and only
if the functions F1(q2) and/or F2(q2) are not constant.” (Drell andZachariasen, 1961, ibid.; emphasis mine)
Invariants & Constitutive a priori in QT, F. Boge 21/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I point particles (Falkenburg, 2007, S. 134 ff.):I (differential) cross section, e.g.
dσdΩ ∝
cos2(θ2 )
sin6(θ2 )
[F2
1 −q2
4M2 [2(F1 + 2MF2)2 tan2(θ2 ) + (2MF2)
2]]
(Rosenbluth; cf. Drell and Zachariasen 1961, p. 8)I Fi = Fi(q2) come from | 〈pout|T|pin〉 |2I “a particle has [...] structure – i.e. is not a point particle – if and only
if the functions F1(q2) and/or F2(q2) are not constant.” (Drell andZachariasen, 1961, ibid.; emphasis mine)
Invariants & Constitutive a priori in QT, F. Boge 21/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I point particles (Falkenburg, 2007, S. 134 ff.):I (differential) cross section, e.g.
dσdΩ ∝
cos2(θ2 )
sin6(θ2 )
[F2
1 −q2
4M2 [2(F1 + 2MF2)2 tan2(θ2 ) + (2MF2)
2]]
(Rosenbluth; cf. Drell and Zachariasen 1961, p. 8)I Fi = Fi(q2) come from | 〈pout|T|pin〉 |2I “a particle has [...] structure – i.e. is not a point particle – if and only
if the functions F1(q2) and/or F2(q2) are not constant.” (Drell andZachariasen, 1961, ibid.; emphasis mine)
Invariants & Constitutive a priori in QT, F. Boge 21/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objective properties of non-isolated objects:I TrE(ρSE) ≈
∑j |αj|
2 |Sj〉〈Sj|
I S =∑
j aj |Sj〉〈Sj| , USES = SUSEI never perfectly applicable, but ‘quasi-classical’:
Invariants & Constitutive a priori in QT, F. Boge 22/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objective properties of non-isolated objects:I TrE(ρSE) ≈
∑j |αj|
2 |Sj〉〈Sj|
I S =∑
j aj |Sj〉〈Sj| , USES = SUSEI never perfectly applicable, but ‘quasi-classical’:
Invariants & Constitutive a priori in QT, F. Boge 22/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objective properties of non-isolated objects:I TrE(ρSE) ≈
∑j |αj|
2 |Sj〉〈Sj|
I S =∑
j aj |Sj〉〈Sj| , USES = SUSEI never perfectly applicable, but ‘quasi-classical’:
Invariants & Constitutive a priori in QT, F. Boge 22/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objective properties of non-isolated objects:I TrE(ρSE) ≈
∑j |αj|
2 |Sj〉〈Sj|
I S =∑
j aj |Sj〉〈Sj| , USES = SUSEI never perfectly applicable, but ‘quasi-classical’:
Invariants & Constitutive a priori in QT, F. Boge 22/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objective properties of non-isolated objects:I TrE(ρSE) ≈
∑j |αj|
2 |Sj〉〈Sj|
I S =∑
j aj |Sj〉〈Sj| , USES = SUSEI never perfectly applicable, but ‘quasi-classical’:
ψ
x
Fψ
p
Invariants & Constitutive a priori in QT, F. Boge 22/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objective correlations:I Uθ( |↑↓〉− |↓↑〉)/
√2 = ( |↑↓〉− |↓↑〉)/
√2 (correlated, regardless of
axis of measurement)I |U( |↑↓〉− |↓↑〉)/
√2|2 = |( |↑↓〉− |↓↑〉)/
√2|2 (probability objective,
not just a degree of belief)I “Correlations have physical reality; that which they correlate does
not.” (Mermin, 1998, p. 753)
Invariants & Constitutive a priori in QT, F. Boge 23/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objective correlations:I Uθ( |↑↓〉− |↓↑〉)/
√2 = ( |↑↓〉− |↓↑〉)/
√2 (correlated, regardless of
axis of measurement)I |U( |↑↓〉− |↓↑〉)/
√2|2 = |( |↑↓〉− |↓↑〉)/
√2|2 (probability objective,
not just a degree of belief)I “Correlations have physical reality; that which they correlate does
not.” (Mermin, 1998, p. 753)
Invariants & Constitutive a priori in QT, F. Boge 23/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objective correlations:I Uθ( |↑↓〉− |↓↑〉)/
√2 = ( |↑↓〉− |↓↑〉)/
√2 (correlated, regardless of
axis of measurement)I |U( |↑↓〉− |↓↑〉)/
√2|2 = |( |↑↓〉− |↓↑〉)/
√2|2 (probability objective,
not just a degree of belief)I “Correlations have physical reality; that which they correlate does
not.” (Mermin, 1998, p. 753)
Invariants & Constitutive a priori in QT, F. Boge 23/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Constructing Quantum Reality
I objective correlations:I Uθ( |↑↓〉− |↓↑〉)/
√2 = ( |↑↓〉− |↓↑〉)/
√2 (correlated, regardless of
axis of measurement)I |U( |↑↓〉− |↓↑〉)/
√2|2 = |( |↑↓〉− |↓↑〉)/
√2|2 (probability objective,
not just a degree of belief)I “Correlations have physical reality; that which they correlate does
not.” (Mermin, 1998, p. 753)
Invariants & Constitutive a priori in QT, F. Boge 23/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Conclusions
I QT can be seen as a guide to expectationsI these expectations cannot easily be viewed as pertaining to the
‘true but unknown’ state λ of some systemI a(n empirical) reality needs to be constructed, by finding out what
the theory objectively prescribesI this may change over time (rotation invariance possibly violated
in q. gravity; cf. Portmann and Wuthrich 2007, p. 849)I correlations may constitute objective part of physical reality while
what is correlated (definitely valued spins etc.) may not
Invariants & Constitutive a priori in QT, F. Boge 24/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Conclusions
I QT can be seen as a guide to expectationsI these expectations cannot easily be viewed as pertaining to the
‘true but unknown’ state λ of some systemI a(n empirical) reality needs to be constructed, by finding out what
the theory objectively prescribesI this may change over time (rotation invariance possibly violated
in q. gravity; cf. Portmann and Wuthrich 2007, p. 849)I correlations may constitute objective part of physical reality while
what is correlated (definitely valued spins etc.) may not
Invariants & Constitutive a priori in QT, F. Boge 24/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Conclusions
I QT can be seen as a guide to expectationsI these expectations cannot easily be viewed as pertaining to the
‘true but unknown’ state λ of some systemI a(n empirical) reality needs to be constructed, by finding out what
the theory objectively prescribesI this may change over time (rotation invariance possibly violated
in q. gravity; cf. Portmann and Wuthrich 2007, p. 849)I correlations may constitute objective part of physical reality while
what is correlated (definitely valued spins etc.) may not
Invariants & Constitutive a priori in QT, F. Boge 24/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Conclusions
I QT can be seen as a guide to expectationsI these expectations cannot easily be viewed as pertaining to the
‘true but unknown’ state λ of some systemI a(n empirical) reality needs to be constructed, by finding out what
the theory objectively prescribesI this may change over time (rotation invariance possibly violated
in q. gravity; cf. Portmann and Wuthrich 2007, p. 849)I correlations may constitute objective part of physical reality while
what is correlated (definitely valued spins etc.) may not
Invariants & Constitutive a priori in QT, F. Boge 24/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Conclusions
I QT can be seen as a guide to expectationsI these expectations cannot easily be viewed as pertaining to the
‘true but unknown’ state λ of some systemI a(n empirical) reality needs to be constructed, by finding out what
the theory objectively prescribesI this may change over time (rotation invariance possibly violated
in q. gravity; cf. Portmann and Wuthrich 2007, p. 849)I correlations may constitute objective part of physical reality while
what is correlated (definitely valued spins etc.) may not
Invariants & Constitutive a priori in QT, F. Boge 24/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
Thank you!
Invariants & Constitutive a priori in QT, F. Boge 25/30
QT in a Nutshell & the MPDecoherence
Human Knowledge?Constructing Reality
ConclusionsReferences
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