The two-state vector formalism (TSVF) is a description of

quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...

in terms of a causal relation in which the present is caused by quantum states of the past and of the future taken in combination.

Theory

The two-state vector formalism is one example of a time-symmetric interpretation of quantum mechanics (see Interpretations of quantum mechanics). Time-symmetric interpretations of quantum mechanics were first suggested by Walter Schottky in 1921, and later by several other scientists. The two-state vector formalism was first developed by Satosi Watanabe in 1955, who named it the Double Inferential state-Vector Formalism (DIVF). Watanabe proposed that information given by forwards evolving

quantum state In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in ...

s is not complete; rather, both forwards and backwards evolving quantum states are required to describe a quantum state: a first state vector that evolves from the initial conditions towards the future, and a second state vector that evolves backwards in time from future boundary conditions. Past and future measurements, taken together, provide complete information about a quantum system. Watanabe's work was later rediscovered by Yakir Aharonov, Peter Bergmann and Joel Lebowitz in 1964, who later renamed it the Two-State Vector Formalism (TSVF). Conventional

prediction A prediction (Latin ''præ-'', "before," and ''dicere'', "to say"), or forecast, is a statement about a future event or data. They are often, but not always, based upon experience or knowledge. There is no universal agreement about the exac ...

, as well as retrodiction, can be obtained formally by separating out the initial conditions (or, conversely, the final conditions) by performing sequences of coherence-destroying operations, thereby cancelling out the influence of the two state vectors. The ''two-state vector'' is represented by: where the state

\langle\Phi,

evolves backwards from the future and the state

, \Psi\rangle

evolves forwards from the past. In the example of the

double-slit experiment In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechani ...

, the first state vector evolves from the electron leaving its source, the second state vector evolves backwards from the final location of the electron on the detection screen, and the combination of forwards and backwards evolving state vectors determines what occurs when the electron passes the slits. The two-state vector formalism provides a time-symmetric description of quantum mechanics, and is constructed such as to be time-reversal invariant. It can be employed in particular for analyzing pre- and post-selected quantum systems. Building on the notion of two-state, Reznik and Aharonov constructed a time-symmetric formulation of quantum mechanics that encompasses probabilistic observables as well as nonprobabilistic weak observables.

Relation to other work

In view of the TSVF approach, and in order to allow information to be obtained about quantum systems that are both pre- and post-selected, Yakir Aharonov, David Albert and Lev Vaidman developed the theory of weak values. In TSVF, causality is time-symmetric; that is, the usual chain of causality is not simply reversed. Rather, TSVF combines causality both from the past (forward causation) and the future (backwards causation, or

retrocausality Retrocausality, or backwards causation, is a concept of cause and effect in which an effect precedes its cause in time and so a later event affects an earlier one. In quantum physics, the distinction between cause and effect is not made at the most ...

). Similarly as the

de Broglie–Bohm theory The de Broglie–Bohm theory, also known as the ''pilot wave theory'', Bohmian mechanics, Bohm's interpretation, and the causal interpretation, is an interpretation of quantum mechanics. In addition to the wavefunction, it also postulates an act ...

, TSVF yields the same predictions as standard quantum mechanics. Lev Vaidman emphasizes that TSVF fits very well with Hugh Everett's many-worlds interpretation, with the difference that initial and final conditions single out one branch of wavefunctions (our world). The two-state vector formalism has similarities with the transactional interpretation of quantum mechanics proposed by

John G. Cramer John Gleason Cramer, Jr. (born October 24, 1934) is a Professor Emeritus of Physics at the University of Washington in Seattle, Washington, known for his development of the Transactional Interpretation of Quantum Mechanics. He has been an activ ...

in 1986, although Ruth Kastner has argued that the two interpretations (Transactional and Two-State Vector) have important differences as well. Avshalom C. Elitzur, Eliahu Cohen: ''The Retrocausal Nature of Quantum Measurement Revealed by Partial and Weak Measurements'', AIP Conf. Proc. 1408: ''Quantum Retrocausation: Theory and Experiment (13–14 June 2011, San Diego, California)'', pp. 120-131,
abstract
It shares the property of time symmetry with the

Wheeler–Feynman absorber theory The Wheeler–Feynman absorber theory (also called the Wheeler–Feynman time-symmetric theory), named after its originators, the physicists Richard Feynman and John Archibald Wheeler, is an interpretation of electrodynamics derived from the ass ...

Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superf ...

and

John Archibald Wheeler John Archibald Wheeler (July 9, 1911April 13, 2008) was an American theoretical physicist. He was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in e ...

and with the time-symmetric theories of Kenneth B. Wharton and Michael B. Heaney
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Theory

Relation to other work

See also

References

Further reading