qitd342 - qitd342 Histories and Consistency Robert B....

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Unformatted text preview: qitd342 Histories and Consistency Robert B. Griffiths Version of 31 January 2012 Contents 1 Histories 1 1.1 Classical stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Quantum histories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.3 Histories sample space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2 Probabilities 5 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Consistency using chain kets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.3 Probabilities using chain kets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 References: CQT = Consistent Quantum Theory by Griffiths (Cambridge, 2002), Chs. 8, 9, 10, 11. 1 H i s t o r i e s 1.1 Classical stochastic processes When probability theory is applied to a sequence of events in time, the sample space consists of what we shall call histories . For example, if a coin is tossed twice in a row, the four histories HH , HT , TH , and TT , where HT stands for “heads on the first toss and tails on the second,” represent mutually exclusive possibilities, one and only one of which will occur in a single experimental run. The event algebra contains 16 “events,” including “heads on the first toss,” “the same both times,” and the like. • The sample space of histories for N tosses of one coin is identical to the sample space of outcomes when N coins are tossed at the same time. The same analogy works for quantum systems. 1.2 Quantum histories Quantum histories consist of a sequence of events at a series of times, but they are now specified by giving a projector on the quantum Hilbert space at each time. Thus for a set of times t 1 < t 2 < · · · t f (1) one can specify a set of projectors F 1 ,F 2 ,... F f , whose significance is that the system of interest has (or had) the property F j at the time t j . 1 • For example, in the case of a qubit we might have F 1 = [0], F 2 = [1], F 3 = [1], where [ ψ ] = | ψ ψ | for a normalized state | ψ . A useful analogy. Consider a composite system consisting of a number of individual systems, where the total Hilbert space is the tensor product of Hilbert spaces for the individual systems: H = H 1 ⊗ H 2 ⊗ ···H f (2) • The projector for a product property of such a composite system can then be written as F = F 1 ⊗ F 2 ⊗ ··· F f . (3) The significance of F is that system 1 has property F 1 , system 2 property F 2 , etc. Following the analogy we define the histories Hilbert space ˘ H = H 1 H 2 · · · H f . (4) ◦ Here indicates a tensor product, the same as ⊗ ; using a distinct symbol in the case of histories is convenient but not essential....
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This note was uploaded on 02/15/2012 for the course PHYS 3101 taught by Professor Staff during the Spring '08 term at Pittsburgh.

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qitd342 - qitd342 Histories and Consistency Robert B....

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