Introduction E9F5FC

Questions FFFFC0

• View
• Edit
• History
• Print

Modeling the Self with Quantum Symmetries of Hamiltonians for Noninteracting Fermions

Interpretations

• {$iJ_1$} and {$iJ_2$} are the relevant operators
• The pseudoscalar {$J_6J_5J_4J_3J_2J_1$} equals the building block for {$J_7$} and for {$J_8$}
• Global constraint, imposing the Hermitian-ness, is love. How to understand the local constraint?
• Qualia are given by the unconscious.
• Division superalgebras extend inaccessibility {$C$} for it encodes {$\mathbb{R}$} ({$C^2=1$}) and {$\mathbb{H}$} ({$C^2=-1$}) and there is also no symmetry which encodes {$\mathbb{C}$}. Whereas accessibility {$T$} encodes {$\mathbb{C}+e\mathbb{C}$} and this {$e$} can be understood as {$T$}.
• Closing the gap (between positive and negative energy levels) changes the insulator by making it into a conductor
• Symmetries are built up from the linear complex structures.
• Walking along the clock, except for the initial interpretation of {$C$}, then we have to build the symmetries from the single operators {$J_k$} (why?) and so first they must be squaring to {$-1$} and so first they must be conscious and only then can the symmetries be unconscious. This is the primacy of the conscious mind for internal experience.
• The diagonal matrices in my octonion interpretation are straightening out the eigenspaces so that they are degenerate.
• The self is modeled by the Hamiltonian.
• The Hamiltonian models the self, thus not the being, but rather what it observes of itself, its personal context as such.
• The perspective is modeled by the linear complex structure {$J_k$}
• The perspective alternatives between {$H$} (stepping in) and {$H^*$} (stepping out) by {$H^*J_k=J_kH$}
• Physics Stack Exchange. Commuting with the Hamiltonian. An operator {$A$} that commutes with the Hamiltonian {$H$} is conserved. In the Heisenberg picture, {$\frac{dA}{dt}=-\frac{i}{\hbar}[A,H]$}. The Hamiltonian establishes the framework for all that is conserved over time. The Hamiltonian is the infinitesimal generator of time translation. Compare with Ehrenfest theorem and {$\frac{d}{dt}\langle A \rangle = \frac{1}{i\hbar}\langle [A,H] \rangle + \langle \frac{\partial A}{\partial t}\rangle$}.
• Commutation of {$A$} and {$H$} means that {$A$} is constant in time. They can both be simultaneously diagonalised. As observables, they can be measured simultaneously. They share the same set of eigenstates and thus any state can be expanded as a sum of these.
• Anticommutation means that {$\frac{dA}{dt}=\frac{2iH}{\hbar}A$} which implies {$A=e^{\frac{2iH}{h}t}$}, the time evolution operator, the propagator, but with time {$t$} reversed.
• From {$HA+AH=0$}, you can directly show that non-zero eigenvalues of the Hamiltonian come in pairs: If {$\psi_n$} is an eigenstate for eigenvalue {$n$}, then {$A\psi_n$} is an eigenstate for {$−n$}. Thus {$A$} takes you from positive eigenstates to negative eigenstates and vice versa.
• If {$A$} is unitary and anticommutes with the Hamiltonian, this implies a symmetry in the energy spectrum: for every eigenstate {$|E\rangle $} of {$H$} with eigenvalue {$E$}, there exists another eigenstate {$A^{-1}|E\rangle$} with energy eigenvalue {$-E$}.
• Balancing prejudices (first mind) and preconceptions (second mind) gives us the absolute frame, the third mind. Letting go of prejudices and preconceptions gets us to the absolute frame by itself.
• Detailed balance - at equilibrium, each elementary process is in equilibrium with its reverse process - is expressed by the geometry of a symmetric space.
• If {$H$} satisfies Schroedinger's equation {$H|\psi(t)\rangle=i\hbar \frac{d}{dt}|\psi(t)\rangle$}, then {$m=-iH$} satisfies {$m|\psi(t)\rangle=-\hbar\frac{d}{dt}|\psi(t)\rangle$} yielding {$|\psi(t)\rangle=e^{-mt/\hbar}$}.
• The measurement yielding an eigenvalue is more a fact about the measurer, how they approached the system, rather than the system measured.
• Unitary symmetries can be
• The presence or absence of anti-unitary symmetries is a matter of reality conditions on the block Hamiltonians (of the Hamiltonians on which the unitary symmetries hold).
• The symmetry of a Hamiltonian indicates that the input states and output states, the creation operators and annihilation operators, the particles and the holes can be swapped. In this sense, the Hamiltonian, the self, is balanced.
• Symmetries {$C$} and {$S$} presume the pairing of particles and holes. But also the distinction is presumed by the {$T$} symmetry {$1_{p,q}H=H1_{p,q}$}. Only the question of when {$p=q$} is undetermined.
• Accessibility (time reversal) moves you back and forth within your world, manifesting two-track attention, thus what is knowable. Inaccessibility (charge conjugation) takes you to another world but in the opposite time direction, thus emphasizing that world's inaccessibility, manifesting a one-track attention, thus what is unknowable. (What do the directions of time mean here? The ability to go back and forth, as with the fivesome.) Definability (sublattice) distinguishes between particles that go to holes and that go to negative holes. Unconsciuos refers to the knower and conscious to the not-knower.
• Knowledge evokes the self. With the fourth perspective, we restrict the self to an eigenspace. The other eigenspace can be identified with the defined self, imposed by sublattice symmetry, and according with it, establishing its basis. Thus knowledge evokes the self and involves a latent perspective bigger than the self, through which the self can be understood.
• Similar to bioelectric fields - surface conduction

Facts

• When {$H_k=J_1m_k$}, we have {$J_1$} and {$J_{k+1}$} commute with {$H_k$}, and {$J_2,\dots,J_k$} anticommute with {$H_k$}.
• When you perform a quantum mechanical measurement, you will always measure an eigenvalue of your operator, and after the measurement your state is left in the corresponding eigenstate. The eigenstates to the operator are precisely those states for which there is no uncertainty in the measurement: You will always measure the eigenvalue, with probability {$1$}. An example are the energy-eigenstates. If you are in a state {$|n\rangle$} with eigenenergy {$E_n$}, you know that {$H|n\rangle=E_n|n\rangle$} and you will always measure this energy {$E_n$}.
• The antilinearity of {$\hat{\mathcal{T}}$}, with {$\hat{\mathcal{T}}\hat{H}\hat{\mathcal{T}}^{-1}=\hat{H}$}, acting on {$\hat{\mathcal{U}}(t)=e^{it\hat{H}}$}, yields time reversal {$\hat{\mathcal{T}}\hat{U}\hat{\mathcal{T}}^{-1}=\hat{U}(-t)$}.
• Polar decomposition of a matrix {$A=UP$} in terms of a unitary matrix {$U$} and a positive semi-definite Hermitian matrix {$P$}.
• Nambu mechanics considers multiple Hamiltonians.
• An anti-symmetric matrix {$M$} (from a Lie algebra) can be decomposed {$M=M_a+M_s$} in terms of an anti-symmetric matrix {$M_a=\frac{1}{2}M-M^T$} (from a Lie subalgebra) and a symmetric matrix {$M_s=\frac{1}{2}M+M^T$}. The symmetric matrix yields, upon exponentiation, the time-evolution operator in the coset space, which is a symmetric space.
• Quantized Hall effect - adding a hole in the insulator
• Time reversal - no magnetism.
• Two classes of time reversal symmetric insulators - in two dimensions: no edge states vs. quantum spin Hall insulator (edge states).
  • Three dimensions: no surface states vs. protected surface conductance
• Can add a crystal reflection symmetry.

Ideas

• The symmetries of context.
• Lurking in the heart of math are the symmetries of math itself.

Sources

• Ben Goertzel. From Complexity to Creativity. Chapter 8. The Structure of Consciousness.