This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
This note was uploaded on 02/15/2012 for the course PHYS 3101 taught by Professor Staff during the Spring '08 term at Pittsburgh.
 Spring '08
 Staff

Click to edit the document details