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Entangled Histories MIT-CTP-4642 Entangled Histories Jordan Cotler1 and Frank Wilczek1;2 1. Center for Theoretical Physics, MIT, Cambridge MA 02139 USA 2. Origins Project, Arizona State University, Tempe AZ 25287 USA February 8, 2015 Abstract We introduce quantum history states and their mathematical framework, thereby reinterpreting and extending the consistent histories approach to quantum theory. Through thought experiments, we demonstrate that our formalism allows us to an- alyze a quantum version of history in which we reconstruct the past by observations. In particular, we can pass from measurements to inferences about \what happened" in a way that is sensible and free of paradox. Our framework allows for a richer under- standing of the temporal structure of quantum theory, and we construct history states that embody peculiar, non-classical correlations in time. Many quantities of physical interest are more naturally expressed in terms of histories than in terms of \observables" in the traditional sense, i.e. operators in Hilbert space that 2 act at a particular time. The accumulated phase exp i dt~v · A~ of a particle moving in ´1 an electromagnetic potential, or its accumulated proper time, are simple examples. We may ask: Having performed a measurement of this more general, history-dependent sort of observable, what have we learned? For conventional observables, the answer is that we learn our system is in a particular subspace of Hilbert space, that is the eigenspace corresponding to the observable's measured value. Here we propose a general framework for formulating and interpreting history-dependent observables. Over the last thirty years, the quantum theory of histories has been approached from several directions. In the 1980's, Griffiths developed a mathematically precise formulation of the Copenhagen interpretation [1]. Griffiths was able to elucidate seemingly paradoxical experiments by enforcing a consistent interpretation of quantum evolution. Omn`es,Gell- Mann, Hartle, Isham, and Linden, among others, enriched the mathematics and physics of Griffith’s theory of \consistent histories" [2]-[11]. In particular, Gell-Mann and Hartle focused on applying consistent histories to decoherence and quantum cosmology, while Isham and Linden's work has uncovered deep mathematical structure at foundations of quantum mechanics [12]. 1 Histories are of course an explicit element of Feynman's path integral. In the early 1990's, Farhi and Gutmann developed a generalized theory of the path integral, which clarifies what is meant by the path integral \trajectories" of a spin system, or any other system with non-classical features [13]. Here we will construct a formal structure that builds on these two lines of work, and il- luminates the question posed in our first paragraph. We will also analyze several instructive examples, where we apply the formalism to analyze quasi-realistic thought experiments. 1 Mathematics of History States 1.1 History Space We will work with a vector space that allows for access to the evolution of a system at multiple times. This vector space is called the history Hilbert space H, and is defined by the tensor product from right to left of the admissible Hilbert spaces of our system at sequential times. Explicitly, for n times t1 < ··· < tn, we have H := Htn · · · Ht1 (1) where Hti is the admissible Hilbert space at time ti. Restricting the admissible Hilbert spaces Ht1 and Htn corresponds to pre- and post-selection respectively. In this paper, we will be primarily concerned with history Hilbert spaces defined over a discrete set of times. It is possible to work with history Hilbert spaces over a continuum of times, but doing so requires the full apparatus of the Farhi-Gutmann path integral [13]. The history Hilbert space H is also equipped with bridging operators T (tj; ti), where T (tj; ti): Hti !Htj . These bridging operators encode unitary time evolution. 1.2 History States We would like to define a mathematical object that encodes the evolution of our system through time { a notion of \quantum state" for history space { that supports an inner product and probability interpretation. One might at first think that an element of H such as j (tn)i · · · j (t1)i would be the desired mathematical object, but a different concept appears more fruitful. For us, \history states" are elements of the linear space Proj(H) spanned by projectors from H!H. Henceforth, we will call Proj(H) the \history state space." For example, if H is the history space of a spin-1/2 particle at three times t1 < t2 < t3, then an example history state is [z−] [x+] [z+] (2) where we use the notation [z+] := jz+ihz+j. The history state in Eqn. (2) can be considered as a quantum trajectory: the particle is spin up in the z-direction at time t1, spin-up in 2 the x-direction at time t2, and spin-down in the z-direction at time t3. Since Proj(H) is a complex vector space, another example of a history state is α [z−] [x+] [z+] + β [z+] [x−] [z+] (3) for complex coefficients α and β. The history state in Eqn. (3) is a superposition of the history states [z−] [x+] [z+] and [z+] [x−] [z+]. It can be interpreted, roughly, as meaning that the particle takes both quantum trajectories, but with different amplitudes. Precise physical interpretation of history states like the one in Eqn. (3) requires more structure. For example, it is not obviously true (and usually is false) that one can measure the particle to take the trajectory [z−] [x+] [z+] with probability proportional to jαj2, or the other trajectory with probability proportional to jβj2. To discuss probabilities, generalizing the Born rule, we need an inner product. Furthermore, we have not yet defined which mathematical objects correspond to measurable quantities. For a history space H with n times t1 < ··· < tn, a general history state takes the form X jΨ) = αi[ai(tn)] · · · [ai(t1)] (4) i where each [ai(tj)] is a one-dimensional projector [ai(tj)] : Htj !Htj , and αi 2 C. We have decorated the history state with a soft ket j · ) which is suggestive of a wave function. In our theory, a history state is the natural generalization of a wave function, and has similar algebraic properties. Note that such sums of products of projectors will also accommodate products of hermitean operators more generally. 1.3 Inner Product In defining a physically appropriate inner product between history states the K operator, defined by X KjΨ) = αiK([ai(tn)] · · · [ai(t1)]) (5) i X = αi[ai(tn)]T (tn; tn−1)[ai(tn−1)] · · · [ai(t2)]T (t2; t1)[ai(t1)] (6) i where T (tj; ti) is the bridging operator associated with the history space, plays a central role. Note that K maps a history state in Proj(H) to an operator which takes Ht1 !Htn . Using the K operator, we equip history states with the positive semi-definite inner product [1] h i (ΦjΨ) := Tr (KjΦ))yKjΨ) (7) This inner product induces a semi-norm on history states, and we call (ΨjΨ) the \weight" of jΨ). It reflects the probability of jΨ) occurring. Note that the inner product is degenerate, in the sense that (ΨjΨ) = 0 does not imply that jΨ) = 0. 3 We say that a history state jΨ) is normalized if (ΨjΨ) = 1. If jΨ) has non-zero weight, then jΨ) jΨ) = (8) p(ΨjΨ) is normalized. We will use this \bar" notation throughout the rest of the paper. At this point, an example may be welcome. If H is the history space of a spin-1/2 particle at three times t1 < t2 < t3 equipped with a trivial bridging operator T = 1, then 1 K [z−] [x+] [z+] = jz−ihz+j (9) 2 Let us interpret the factor of 1=2 on the right-hand side of the above equation. Since the bridging operator for the history space is the identity, any particle which is spin-up in the z-direction at time t1 will continue to be in that state at times t2 and t3. However, the history state in Equation (9) is [z−] [x+] [z+] which appears to subvert the unitary evolution imposed by the bridging operator. This subversion comes at a cost, which is the amplitude 1=2. We will later see that histories which do not follow unitary evolution have a suppressed probability of being measured, with the suppression factor proportional to the absolute square of the coefficient generated by the K operator. In the case of Equation (9), the \suppression" is a factor of j1=2j2 = 1=4. 1.4 Families We will be interested in subspaces that both admit an orthogonal basis (possibly including history states of zero norm), and contain the history state 1tn · · · 1t1 , which corresponds to a system being in a superposition of all possible states at each time. The orthogonal set of history states which spans such a subspace will be called a \family" of history states. More formally: i Definition We say fjY )g is a family of history states if i j i i (1) (Y jY ) = 0 for i 6= j and (Y jY ) = 0 or 1, and P i (2) i ci jY ) = 1tn · · · 1t1 for some complex ci. Requirement (1) is Griffiths’ \strong consistent histories condition" [1]-[3]. Contrary to earlier work, we see no reason to impose the requirement that history states, regarded as operators, commute. It is not essential that history states commute i i since they are not themselves observables. However, projectors of the form jY )(Y j are i i j j observables. By orthogonality, [jY )(Y j; jY )(Y j] = 0 for all i; j, so commutativity of the corresponding observables is automatic.
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