In this note I indicate how the local transformation gauge-like physics of
gravitation can preserve the unitarity of a quantum system, In this there
is an identification of the conformal group of spacetime physics or general
relativity with an orthogonal gauge group from the Bloch vector, or
extended analogue thereof, with the gauge-like SU(2,2) group of conformal
gravitation. A central feature of this is this gauge-like theory conserves
the entanglement structure of an associated orthogonal group or with SU(2,
2) ↔ O(15), or to take U(2,2) = SU(2,2)×U(1) we have U(2,2) ↔ SO(16)
A central aspect in the formalism of entanglement is the Schmidt
decomposition. A general tensor product of states is
|ψ⟩ = sum_{ij}C_{ij}|i⟩⊗|j⟩.
Now consider the intermediary state |φ_i⟩ = sum_jC_{ij}|j⟩, which forms |ψ⟩
= sum_i|i⟩⊗|φ_i⟩. The density matrix ρ = |ψ⟩⟨ψ| has the trace with respect
to the φ states
Tr_φ(ρ) = sum_{ij} ⟨φ_j |φ_i⟩ |i⟩⟨j|,
and we have ⟨φ_j |φ_i⟩ = p_iδ_{ij}. Clearly then we can write
|ψ⟩ = sum_{ij}√(p_i)|i⟩⊗|j⟩.
This is also a polar form of the wave function.
This Schmidt form is with
√(p_i) = sum_jC_{ij}
where this amplitude transforms as
C_{ij} → C’_{ij} = U_{ik}√(p_k)V_{kj},
The two unitary matrices U_{ik} and V_{kj} are transformations of the two
states |i⟩ and |j⟩ and the transformation of the amplitude C_{ij} is given
by the product. A simple case is with a simple spin model of SU(2) so the
product of the two unitary groups is SU(2)×SU(2) = SO(4).
We can use this with density matrices as a set of diagonal plus
off-diagonal states. Consider, the N×N density matrix
ρ_{ij} = p_iδ_{ij} + σ_iτ_j, i and j =1 … N^2 - 1
where the first term corresponds to maximally mixed states for p_i = 1/N
and the second terms are the off-diagonal quantum phase. Here σ_i is a
generator of an SU(N) group and τ_j a B;och vector. The space of all unit
trace Tr(ρ) = 1 of all N×N density matrices is a manifold designated by ℳ.
This manifold is the intersection, in the set of all Hermitian matrices, of
a positive cone P with the hyperplane parallel to all linear traceless
operators. This is a convex set defined by the set of all projectors onto a
one-dimensional subspace. This the defines the projective geometry ℂP^{N-1}
in the Hilbert space of ℳ.
There is more geometry I could discuss but will defer to later. This
involves some subtle issues with the relationship between the diagonal
trace terms and the off diagonal term corresponding to quantum phases.
The group theoretic implications of this are then interesting. Consider the
rotation of the Bloch vector τ_i → τ’_i according the unitary
transformation of the density matriz
τ’_i = ½Tr(ρ’)σ_i = [ σ_kU_{ki}σ_lU^†_{lj} ] τ_j = sum_jO_{ij} τ_j.
We then have SO(N^2 - 1) matrices that are associated with SU(N), or more
properly that SU(N)/ℤ_N is a subset of SO(N^2 - 1). To consider this let N
= 4, then we have that SU(4)/ ℤ_4 is a subset of SO(15). SO(15) is fixed in
a frame of SO(16) and this corresponds to a U(1) fibration over SU(4) as
U(4) = SU(4)×U(1).
The Hopf fibration defines the sphere S^4 = O(6)/O(5) or that O(6) ≈ U(4)
is the 4-sphere with an O(5) fibration. If we shift to a hyperbolic setting
then we have O(4,2)/SO(5,1) = AdS_5. with the quotient on the O(4,2) =
U(2,2). We then clearly have a correspondence with the orthogonal group
SO(16). The correspondence to AdS_5×S^5 is then with the Cartan
decomposition SO(32) → SO(16)×*120* and the corresponding unitary group is
U(2,2,ℂ) in complex conformal relativity. There is a conservation of
information between the U(2,2) and SO(16), where the first pertains to
conformal gravitation and the latter a gauge field theory.
There is the issue of a “sleight of hand” where the unitary group is in
split form, corresponding to spacetime with Lorentzian metric and the
orthogonal group is Euclidean and corresponds to a gauge group. The claim
here is that for the Lorentzian group a difficulty is this leads to
negative probabilities. However, this really is not as bad as one might
think. Coherent states, such as with laser photons or condensates, have
this feature. These forms of quantum states have both a Riemannian and
symplectic geometric structure. These over-complete quantum states give a
way that classical-like structure can emerge from quantum physics. The
central feature of pure state quantum mechanics is linearity of Hilbert
space of states and operators. The transition to nonlinearity with this
conservation of information, say qubits ↔ spatial or spacetime information,
is a feature of how state collapse and the stability of classical states
does not violate conservation of information.
Noq consider the group theoretic implications of this are then interesting.
Consider the rotation of the Bloch vector τ_i → τ’_i according the unitary
transformation of the density matriz
τ’_i = ½Tr(ρ’)σ_i = [ σ_kU_{ki}σ_lU^†_{lj} ] τ_j = sum_jO_{ij} τ_j.
which we rewrite for the entanglement of states
|ψ⟩ = sum_{ij}C_{ij}|i⟩⊗|j⟩
So the amplitude with Schmidt form √(p_i) = sum_jC_{ij} transforms as
C_{ij} → C’_{ij} = U_{ik}√(p_k)V_{kj},
The two unitary matrices U_{ik} and V_{kj} are transformations of the two
states |i⟩ and |j⟩ and the transformation of the amplitude C_{ij} is given
by the product. The density matrix is then expanded as
ρ_{ij} = p_iδ_{ij} + σ_iτ_j, i and j =1 … N^2 - 1
and the Bloch vector with τ’_iτ_j = ½Tr(ρ’)σ_iτ_j = ½Tr(ρ’)g_{ij} is also
τ’_iτ_j = sum_kO_{ik}τ_kτ_j. This is a form of metric.
The entangled state defines
|ψ⟩⟨ψ| = sum_{ij}C*_{ij}C_{ij}|i⟩⊗|i⟩⟨j|⊗⟨j|.
such that the transformation of the amplitude square is
C’*_{ij} C’_{ij} = sum_{kk’}√(p_kp_k’)U_{ik}V_{kj} U_{ik’}V_{k’j},
So, the density matrix is identified as
ρ_{ij} = sum_{kk’}O_{ik}O’_{k’j} τ_kτ_k’
where the O_{ik} is the SO(N^2 - 1) transformation with U and O_{k’j} is
the SO(N^2 – 1) transformation with V. This is then a form with a metric on
the Bloch sphere given by τ_kτ_k’.
The entangled state with C_{ij} = e^{-E_{ij}β/2}, ordinarily β = 1/kT in
Euclidean form, but equivalent to it/ħ in Lorentzian form, can be used to
examine the overlap of states. We compute to O(δt^3) the overlap ⟨ψ|ψ +
δψ⟩, here with the variation with respect to proper time in a local frame,
⟨ψ|ψ + δψ⟩ = 1 - (i/ħ)⟨H⟩δt - (1/2ħ^2)(⟨H^2⟩ - iħ)⟨H’⟩)δt^2
+ (1/3!ħ^3)(i⟨H^3⟩ - iħ^2⟨H’’⟩)δt^3
The derivative H’ of the Hamiltonian H = sum_{i=0}^{N^2-1}ε_iσ_i is
determined by the covariant derivative of σ_i
Dσ_i/ds = u∇_uσ_i = u∂_uV(σ_i’) + uΓV(σ_i’),
which corresponds to a geodesic motion and is then zero for V = u. The
cryptic term (σ_i’) just means a differential of the bloch vector with
respect to spatial or spacetime coordinates and V = dx/ds. The second
derivative H’’ is determined by
D^2σ_i/ds^2 = d/ds(∇σ)u = σ_i’R(u,V,u) → (e_i’)_μR^μ_{ανβ}U^αV^νU^β,
which is a geodesic deviation equation. The vector σ_i is determined by a
vector connecting two regions of spacetime. These contribute to O(δt) and
δt^3. However, when we compute the modulus square, we find
|⟨ψ|ψ + δψ⟩|^2 = 1 - (1/ħ^2)(⟨H^2⟩ - ⟨H⟩^2)δt^2 .
where the geodesic condition removes any ⟨H’⟩ and the geodesic motion has
not contribution because the δt^3 is imaginary valued and subtracts out.
This recovers the standard measure for quantum metric distance with ΔH =
⟨H^2⟩ - ⟨H⟩^2 and this gives the Heisenberg uncertainty principle.
This illustrates how gravitation can maintain unitarity as a sort of
equivalence principle. The dynamics with the vector σ_i defines the
transformation of the matrices O_{ij} that transform the Bloch vectors. The
violation of unitarity occurs though at higher orders. However, if we
consider the variation in the time or proper time as related to a variation
in the spacetime, then δt, δt^2 and δt^3 define a fluctuation in a linear
direction, an area and a volume. The dynamics of spacetime is according to
the dynamics of space with respect to time. Holography tells us as well
that δL^3/L^3 = ℓ_p^2/L^2 so that δL^3 = ℓ_p^2L is the highest order term
that is physically relevant.
It is then apparent the equivalence principle expressed as the invariance
of an entanglement between two particles in curved spacetime preserves
unitarity. The modulus square |⟨ψ|ψ + δψ⟩|^2 = 1 - (1/ħ^2)(⟨H^2⟩ -
⟨H⟩^2)δt^2 is the Fubini-Study metric for QM according to the projective
fibration π:ℋ → Pℋ which for a finite Hilbert space is ℂP^n. In this way
the quantum phase structure of this internal gauge group associated with
conformal group of relativity is conserved.
LC
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