Density matrix. It seems very strange and uncomfortable that our fundamental equation is a linear equation for the wavefunction while all observed quantities involve bi-linear combinations of the wavefunc- tion. This disparity is also at the heart of the measurement paradox because when we measure our interpretation is based on bi-linear constructions, not to mention that people often talk of the wave- function collapse. This is just unscientific because wavefunctions are subject to the Schr¨odinger Equation which does not have a 'collapse' term in it! In what follows I will discuss the density matrix approach to the same foundational problems/questions, so do not be surprised that the same issues will be discussed again but from a different "angle". Let us introduce the notion of the density matrix, which offers an alternative and more powerful (for open systems when we need to deal with the "rest of the world") description of system states. For a system isolated from its environment we define it as (below x is a collection of coordinates, or basis state index, in whatever basis, in our system) 0 ∗ ρ = j i h j ! ρx0x = (x ) (x) : (1) P or in any other basis, after substituting j i = n njni, ∗ hnjρ^jmi = ρnm = n m : (2) We immediately see thatρ ^ is a Hermitian operator with non-negative diagonal elements, and for P normalized states n ρnn = Tr ρ^ = 1. This allows one to interpret ρnn as the probability of finding a system in the state jni. Of course, a matrix parameterized by a vector is a special case; we will refer to this situation as a "pure" state. We will see shortly that for open systems this will no longer be the case, in general, and then we are dealing with a "mixed" state of the density matrix (see below). Its time dependence immediately follows from the Schr¨odinger Equation : @ i ρ(x0; x; t) = jH^ (x0; t)ih (x; t)|−| (x0; t)ihH^ (x; t)j = H^ j (x0; t)ih (x; t)|−| (x0; t)ih (x; t)jH;^ ~ @t (3) where in the last relation I have used the fact that H^ is Hermitian. In matrix notation we have @ i ρ^ (t) = H^ ρ^ (t) − ρ^ (t)H^ = [H;^ ρ^] ; (4) ~ @t nm nk km nk km nm which can be formulated in the basis independent form @ i ρ^(t) = [H;^ ρ^] : (5) ~ @t For time-independent Hamiltonians with eigenstates/eigenenergies fjni;Eng, we immediately get after trivial integration that in the jni-basis it(Em−En)=~ ρnm(t) = ρnm(0)e : (6) This formula is especially important for discussion of the apparent de-phasing relaxation in a system isolated with extreme degree of accuracy. If, after a long enough time of system's evolution, we 1 completely loose control on the value of the phases t(Em − En)=~ modulo 2π, then, FAPP we 1Due to natural limitations on experimental resolution and degree of isolation. 1 should replace our actual density matrix with the one averaged over the values of the phases t(Em − En)=~. This amounts to vanishing of the off-diagonal terms: ρnm(t) ! δnmρnn(0); (7) meaning that our state is indistinguishable from a statistical mixture of energy eigenstates, the state jni entering the ensemble with the probability ρnn(0). [In the degenerate subspaces of H, one can always make H and ρ diagonal at the same time by basis rotations]. Measurement results are also directly related to the density matrix because X ∗ ^ X ^ ^ hΛi = λΛλλ λ = ρ^λλΛλλ = Tr ρ^Λ : (8) λ λ Note that any basis can be used to do the calculation using last equality. Measurement We see that all the necessary ingredients for doing quantum mechanical calculations can be formulated at the level of density matrix. We still need the notion of the Hilbert space and the basis states of the Hermitian operators, but this is basically for mathematical convenience. Of course, dealing with vectors=wavefunctions is easier than with matrixes. But if we are after the physics principle, then density matrix should be our prime object for philosophical discussions because only ρ^ can deal with open system. Why do we need open systems? Because truly isolated ones cannot be observed from outside (!) and thus are irrelevant for material world. Any event qualifying for "system's observation" necessarily involves system's interaction with the outside world, or environment, and has to deal with enlarged Hilbert space of the system and "observer". However, in system's measurements we do not care what happens to the rest of the world (otherwise we are measuring more than the system in question), and thus probabilities for all possible outcomes for the world variables have to be summed up. This leads to the following definition of the density matrix for an open system: Ifρ ^(F ) is a full density matrix for the system and its environment then the system's density matrix is defined as (S) (F ) ρ^ = Trenv ρ^ ; (9) where the trace is taken over the basis states of the environment, or, using a notation Renv for the collection of all degrees of freedom in the environment Z (S) 0 0 ∗ ρ^ (x ; x) = dRenv Ψ(x ; Renv)Ψ (x; Renv) ; env which is an overlap of environmental states for different system parameters. Indeed, the average (S) of any system's operator hΨjQ^jΨi reduces to TrS ρ^ Q^ because Q^ is not acting on environmental coordinates. Now consider the measuring process of some system's quantity characterized by operator Λ^ (with the set fλg of possible measurement outcomes) as a physical process involving sufficiently strong coupling (interaction Hamiltonian) between the system and the environment=measuring machine, something similar to the SG-machine. The set of Λ^ eigenvectors jeλi is very convenient 2 because it is tuned to the properties of the measuring machine in such a way that when our system is in state jeλi the machine is responding with its degrees of freedom strongly enough and goes into the state jMλi such that an FAPP principle by Bell becomes applicable, namely the overlap 0 hMλjMλ0 i is an FAPP zero for λ 6= λ . This is because there are macroscopically many degrees of freedom which change their states strongly and differently enough for λ 6= λ0 under the action P ^ of the interaction Hamiltonian HΛ;env = λ jλihλj henv(λ). In the simplest example it can be a steel arrow of the voltmeter moving to a particular position. If this is not enough, complement it with your journal recording. Consider an arbitrary initial state expanded into the jeλi-basis (for simplicity, we will assume that λ is non-degenerate; more general formalism will be introduced later) X jΨi = λ(Renv)jeλi : λ To ensure that before the measurement the environment is ”indifferent" to the state of the system, we write the expansion coefficients as a product λ(Renv) = Φ(Renv) λ; i.e. the initial state is a P simple product of the system and environment wavefunctions, jΨi = jΦi λ λjeλi. It will evolve during the measurement process into (the process has to be fast with respect to the system's dynamics to qualify for the "measurement of Λ in a given state of the system") X −i[H^env+h^env(λ)]t X −! Ψ = e Φ(Renv) λjeλi = Mλ(Renv) λjeλi : λ λ Correspondingly, (S) X ∗ ρ^ = λ0 λ jeλi heλ0 j : λλ0 will evolve into (S) X ∗ X ∗ X 2 ρ^ = hMλ0 jMλi λ0 λ jeλi heλ0 j = δλλ0 λ0 λ jeλi heλ0 j = j λj jeλi heλj ; (10) λλ0 λλ0 λ with hMλjMλ0 i = δλλ0 . This density matrix is diagonal in the λ-representation and leads to X (S) X hΛi = ρλλ λ = pλλ : (11) λ λ The subsequent evolution of this density matrix is exactly the same as the evolution of an ensemble of independent system's copies each in one of the eigenstates jeλi, as if each copy has collapsed into the state jeλi because the observed value of Λ^ happened to be λ. If λ values have degeneracy ν, then the result will be (S) X ∗ ρ^ = λν λν0 jeλνi heλν0 j ; (12) λ,ν,ν0 or, identically, ∗ (S) X X λν λν0 ρ^ = pλ jeλνi heλν0 j ; (13) pλ λ ν,ν0 with X 2 pλ = j λνj : (14) ν 3 This is essentially a 'multiple Universe' or 'multiverse' interpretation of Quantum Mechanics: measuring process splits the world into distinctive states with FAPP zero overlap; these states FAPP never interfere in the future and go their separate ways FAPP forever. Please, do not worry about those multiple splittings and where they are all 'stored' - the Hilbert space of 1060 degrees of freedom is FAPP infinite! Mathematical formalism Since splitting things into the system and environment is rather arbitrary and mostly depends on the properties of the measured quantity Λ, let us summarize our discussion into the follow- ing formal rules/procedures. When quantity Λ is measured, the possible outcomes are given by probabilities pλ = Tr ρ^P^λ; (15) where P^λ is a projector on the subspace spanned by eigenstates jeλ,νi with the same eigenvector X P^λ = jeλ,νiheλ,νj : (16) ν Here index ν refers to all possible degeneracies for a given eigenstate of Λ or degrees of freedom decoupled from Λ (generalizing previous discussion). FAPP the new density matrix is given by ^ ^ ^ ^ (new) Pλρ^Pλ Pλρ^Pλ X ρ^λ,ν;λ,ν0 ρ^ = = = jeλ,νi heλ,ν0 j : (17) ^ ^ pλ pλ Tr Pλρ^Pλ νν0 The denominator in (17) is a normalization constant necessary to obey Trρ ^(new) = 1.