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Density Matrices as Wave Functions

Given a wave function psi(x,t), the [pure] density matrix is defined as rho(x,x',t) = psi*(x,t) psi(x',t). Taking two copies of the wave function cancels the arbitrary phase.

Density Matrix Formalism Class notes by Frank Porter uses bra-ket notation.

Density Operator chapter in Quantum Optics book by S. Kryszewski





Density Matrices as Spin Functions or Qubits

For spin-1/2 systems suitable for non relativistic electrons, the density matrix becomes a 2x2 matrix. Matrices can also be thought of as a reduction in the number of degrees of freedom of a density matrix wave function.

Density Operator of a Single Qubite: The Bloch Sphere  Density operators in quantum computing. Also see The Measurement the longer chapter on density matrices where the above is taken from.

Introduction to Quantum Mechanics through Density Matrices  by Scott Aaronson. Focuses on qubit theory, mixed states, generalization of probabilities to negative numbers.

MUBs, nilpotents and idempotents of Clifford algebras by Carl Brannen Pure spin density matrices are idempotents, that is, such a matrix B satisfies BB = B. Quantum field theory uses Grassmann variables that are nilpotent so BB = 0. The relationship between nilpotents and idempotents on a Clifford algebra are discussed.

Berry phase and the U(1) gauge symmetry by Carl Brannen The U(1) gauge symmetry of quantum mechanics can be derived by assuming that any quantum system represented by a wave function can also be represented by a density matrix.

Berry ( or Pancharatnam-Berry or quantum ) phase by Carl Brannen Berry phase is the phase that a quantum system picks up when its spin vector is sent through a series of directions. This can be expressed in pure density matrices. The phase picked up is proportional to the surface of the area cut out of the Bloch sphere.

Quantum Electrodynamics of qubits  An introduction to the use of Feynman diagrams in qubit theory. Propagators as density matrices or projection operators.

The Measurement Algebra Julian Schwinger's measurement algebra is a formalism for quantum mechanics / quantum field theory that can be thought of as pure density matrices. The formalism looks at the process of measurement as central and centers on the Sturm-Gerlach experiment, the action of which on a beam of elementary particles can be described by the density matrix form of the spins of the elementary particles.

Advanced Theory of Density Matrices

Consistent Histories and Density Operator Formalism by Carl Brannen The Consistent Histories interpretation of quantum mechanics is written elegantly in pure density matrices. This gives an interpretation of Margaret Hawton's photon position operator

On the Role of Density Matrices in Bohmian Mechanics Conditional density matrices: conditional on the state of the environment. Interpreted using Bohmian mechanics. Spin density matrices. Also see Bohmian Mechanics and Quantum Field Theory, quant-ph/0303156 by same authors.

The Real Density Matrix Timoty Havel, MIT quantum information, a slide show. Defines a new matrix multiplication that allows n-qubit states to be defined without the use of the imaginary number i. Replaces Hermitian density matrices with non symmetric real matrices. Also see The Real Density Matrix quant-ph/0302176. Basically, the idea is similar to the replacement of 4x4 Hermitian matrices with Dirac bilinears; the coefficients are real.

Quantum Bound States: the Hydrogen Atom by Carl Brannen The algebra of density matrices, low dimensional Hilbert spaces.

Quantum States as Symmetry Operators As opposed to spinors, matrices are operators and can operate on matrices. Therefore spin density matrices, or quantum states, can be thought of as operators on quantum states. Quantum states as symmetry operators acting on quantum states.

Non Hermitian Density Matrices Products of two different spin density matrices are non Hermitian. Working with spin density matrices as propagators, these can be interpreted as the transformation of a particle from one state to another, a creation-annihilation pair.

Bound States as Density Matrices Representing the valence quarks of a baryon with density matrices, non Hermitian density matrices correspond to the gluon interactions. The bound states can be put into matrix form and satisfy the usual idempotency equations of a density matrix.

Bound States as Symmetry Operators and E8 by Carl Brannen Matrices of non Hermitian density matrices acting as symmetry operators and interpreted as bound states, relationship to E8 Lie group and algebra.

Broken E8 as a Result of Composite Particles I Since bound states can be represented by density matrices that define symmetries on the constitutent states and on themselves, one ends up with a tower of composite states. When the symmetries of the states are exhausted, one can add no more states. The symmetries on the bound states are defined by the algebra of the bound states. This defines the symmetry group E8 which is most simply described as "the group of symmetries of its own algebra."

The Quantum Zeno Effect The quantum Zeno effect disappears when fully analyzed using density matrices.

Quarks, leptons and generations! The generation structure of the quarks and leptons arranged in 3x3 matrices. Triality with the E8 group.




Applications of Density Matrices

Density operators for spin-1/2 ensembles  Dissipation and dephasing, the Bloch equations. Stanford class Ph195.


"Density Matrix Renormalization: A Review of the Method and its Applications" cond-mat/0303557 by Karen Hallberg A brief review of density matrix renormalization with a concentration on applications in condensed matter physics, statistical mechanics and high energy physics.

"Applications of the density matrix renormalisation group to problems in magnetism" cond-mat/9608127 by Gehring, Bursill, Xiang White's density matrix renormalization group with applications to frustrated spin systems, quantum critical phenomena, 1-d systems at nonzero temperature, and low energy 2-d systems at low energy.

"Density Matrix and Renormalization for Classical Lattice Models" cond-mat/9610107 by Nishino, Okunishi Review of density matrix renormalization group with variational principle