Retrocausality

Last updated

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. [1] [2] In quantum physics, the distinction between cause and effect is not made at the most fundamental level and so time-symmetric systems can be viewed as causal or retrocausal. [3] [ page needed ] Philosophical considerations of time travel often address the same issues as retrocausality, as do treatments of the subject in fiction, but the two phenomena are distinct. [1]

Contents

Philosophy

Philosophical efforts to understand causality extend back at least to Aristotle's discussions of the four causes. It was long considered that an effect preceding its cause is an inherent self-contradiction because, as 18th century philosopher David Hume discussed, when examining two related events, the cause is by definition the one that precedes the effect. [4] [ page needed ]

The idea of retrocausality is also found in Indian philosophy. It was defended by at least two Indian Buddhist philosophers, Prajñākaragupta (ca. 8th–9th century) and Jitāri (ca. 940–1000), the latter wrote a specific treatise on the topic, the Treatise on Future Cause (Bhāvikāraṇavāda). [5] In the 1950s, Michael Dummett wrote in opposition to such definitions, stating that there was no philosophical objection to effects preceding their causes. [6] This argument was rebutted by fellow philosopher Antony Flew and, later, by Max Black. [6] Black's "bilking argument" held that retrocausality is impossible because the observer of an effect could act to prevent its future cause from ever occurring. [7] A more complex discussion of how free will relates to the issues Black raised is summarized by Newcomb's paradox. Essentialist philosophers have proposed other theories, such as the existence of "genuine causal powers in nature" or by raising concerns about the role of induction in theories of causality. [8] [ page needed ] [9] [ page needed ]

Physics

Most physical theories are time symmetric: microscopic models like Newton's laws or electromagnetism have no inherent direction of time. The "arrow of time" that distinguishes cause and effect must have another origin. [10] :116 To reduce confusion, physicists distinguish strong (macroscopic) from weak (microscopic) causality. [11]

Macroscopic causality

The imaginary ability to affect the past is sometimes taken to suggest that causes could be negated by their own effects, creating a logical contradiction such as the grandfather paradox. [12] This contradiction is not necessarily inherent to retrocausality or time travel; by limiting the initial conditions of time travel with consistency constraints, such paradoxes and others are avoided. [13]

Aspects of modern physics, such as the hypothetical tachyon particle and certain time-independent aspects of quantum mechanics, may allow particles or information to travel backward in time. Logical objections to macroscopic time travel may not necessarily prevent retrocausality at other scales of interaction. [14] [ page needed ] Even if such effects are possible, however, they may not be capable of producing effects different from those that would have resulted from normal causal relationships. [15] [ page needed ]

Physicist John G. Cramer has explored various proposed methods for nonlocal or retrocausal quantum communication and found them all flawed and, consistent with the no-communication theorem, unable to transmit nonlocal signals. [16]

Relativity

"In relativity, time and space are intertwined in the fabric of space-time, so time can contract and stretch under the influence of gravity." [17] Closed timelike curves (CTCs), sometimes referred to as time loops, [17] in which the world line of an object returns to its origin, arise from some exact solutions to the Einstein field equation. However, the chronology protection conjecture of Stephen Hawking suggests that any such closed timelike curve would be destroyed before it could be used. [18] Although CTCs do not appear to exist under normal conditions, extreme environments of spacetime, such as a traversable wormhole or the region near certain cosmic strings, may allow their brief formation, implying a theoretical possibility of retrocausality.[ citation needed ] The exotic matter or topological defects required for the creation of those environments have not been observed. [19] [ page needed ] [20] [ page needed ]

Microscopic causality

Most physical models are time symmetric; [10] :116 some use retrocausality at the microscopic level.

Electromagnetism

Wheeler–Feynman absorber theory, proposed by John Archibald Wheeler and Richard Feynman, uses retrocausality and a temporal form of destructive interference to explain the absence of a type of converging concentric wave suggested by certain solutions to Maxwell's equations. [21] These advanced waves have nothing to do with cause and effect: they are simply a different mathematical way to describe normal waves. The reason they were proposed is that a charged particle would not have to act on itself, which, in normal classical electromagnetism, leads to an infinite self-force. [21]

Quantum physics

Time runs left to right in this Feynman diagram of electron-positron annihilation. When interpreted to include retrocausality, the electron (marked e ) was not destroyed, instead becoming the positron (e ) and moving backward in time. Feynmann Diagram Gluon Radiation.svg
Time runs left to right in this Feynman diagram of electron–positron annihilation. When interpreted to include retrocausality, the electron (marked e ) was not destroyed, instead becoming the positron (e ) and moving backward in time.

Ernst Stueckelberg, and later Richard Feynman, proposed an interpretation of the positron as an electron moving backward in time, reinterpreting the negative-energy solutions of the Dirac equation. Electrons moving backward in time would have a positive electric charge. [22] This time-reversal of anti-particles is required in modern quantum field theory, and is for example a component of how nucleons in atoms are held together with the nuclear force, via exchange of virtual mesons such as the pion. A meson is made up by an equal number of normal quarks and anti-quarks, and is thus simultaneously both emitted and absorbed. [23]

Wheeler invoked this time-reversal concept to explain the identical properties shared by all electrons, suggesting that "they are all the same electron" with a complex, self-intersecting world line. [24] Yoichiro Nambu later applied it to all production and annihilation of particle-antiparticle pairs, stating that "the eventual creation and annihilation of pairs that may occur now and then is no creation or annihilation, but only a change of direction of moving particles, from past to future, or from future to past." [25] The backwards-in-time point of view is nowadays accepted as completely equivalent to other pictures, [26] but it has nothing to do with the macroscopic terms "cause" and "effect", which do not appear in a microscopic physical description.

Retrocausality is associated with the Double Inferential state-Vector Formalism (DIVF), later known as the two-state vector formalism (TSVF) in quantum mechanics, where the present is characterised by quantum states of the past and the future taken in combination. [27] [28]

Retrocausality is sometimes associated with nonlocal correlations that generically arise from quantum entanglement, including for example the delayed choice quantum eraser. [29] [30] However accounts of quantum entanglement can be given which do not involve retrocausality. They treat the experiments demonstrating these correlations as being described from different reference frames that disagree on which measurement is a "cause" versus an "effect", as necessary to be consistent with special relativity. [31] [32] That is to say, the choice of which event is the cause and which the effect is not absolute but is relative to the observer. The description of such nonlocal quantum entanglements can be described in a way that is free of retrocausality if the states of the system are considered. [33]

Tachyon visualization: since a tachyon moves faster than the speed of light, we can not see it approaching. After a tachyon has passed nearby, we would be able to see two images of it, appearing and departing in opposite directions. The black line is the shock wave of Cherenkov radiation, shown only in one moment of time. Tachyon04s.gif
Tachyon visualization: since a tachyon moves faster than the speed of light, we can not see it approaching. After a tachyon has passed nearby, we would be able to see two images of it, appearing and departing in opposite directions. The black line is the shock wave of Cherenkov radiation, shown only in one moment of time.

Tachyons

Hypothetical superluminal particles called tachyons have a spacelike trajectory, and thus can appear to move backward in time, according to an observer in a conventional reference frame. Despite frequent depiction in science fiction as a method to send messages back in time, hypothetical tachyons do not interact with normal tardyonic matter in a way that would violate standard causality. Specifically, the Feinberg reinterpretation principle means that ordinary matter cannot be used to make a tachyon detector capable of receiving information. [34]

Parapsychology

Retrocausality is claimed to occur in some psychic phenomena such as precognition. J. W. Dunne's 1927 book An Experiment with Time studied precognitive dreams and has become a definitive classic. [35] Parapsychologist J. B. Rhine and colleagues made intensive investigations during the mid-twentieth century. His successor Helmut Schmidt presented quantum mechanical justifications for retrocausality, eventually claiming that experiments had demonstrated the ability to manipulate radioactive decay through retrocausal psychokinesis. [36] [37] Such results and their underlying theories have been rejected by the mainstream scientific community and are widely accepted as pseudoscience, although they continue to have some support from fringe science sources. [38] [ page needed ] [39] [ page needed ] [40] [ unreliable source? ]

Efforts to associate retrocausality with prayer healing have been similarly rejected. [41] [42]

From 1994, psychologist Daryl J. Bem has argued for precognition. He subsequently showed experimental subjects two sets of curtains and instructed them to guess which one had a picture behind it, but did not display the picture behind the curtain until after the subject made their guess. Some results showed a higher margin of success (p. 17) for a subset of erotic images, with subjects who identified as "stimulus-seeking" in the pre-screening questionnaire scoring even higher. However, like his predecessors, his methodology has been strongly criticised and his results discounted. [43]

See also

Related Research Articles

<span class="mw-page-title-main">Einstein–Podolsky–Rosen paradox</span> Historical critique of quantum mechanics

The Einstein–Podolsky–Rosen (EPR) paradox is a thought experiment proposed by physicists Albert Einstein, Boris Podolsky and Nathan Rosen which argues that the description of physical reality provided by quantum mechanics is incomplete. In a 1935 paper titled "Can Quantum-Mechanical Description of Physical Reality be Considered Complete?", they argued for the existence of "elements of reality" that were not part of quantum theory, and speculated that it should be possible to construct a theory containing these hidden variables. Resolutions of the paradox have important implications for the interpretation of quantum mechanics.

Faster-than-light travel and communication are the conjectural propagation of matter or information faster than the speed of light. The special theory of relativity implies that only particles with zero rest mass may travel at the speed of light, and that nothing may travel faster.

<span class="mw-page-title-main">Quantum entanglement</span> Correlation between quantum systems

Quantum entanglement is the phenomenon of a group of particles being generated, interacting, or sharing spatial proximity in such a way that the quantum state of each particle of the group cannot be described independently of the state of the others, including when the particles are separated by a large distance. The topic of quantum entanglement is at the heart of the disparity between classical and quantum physics: entanglement is a primary feature of quantum mechanics not present in classical mechanics.

A tachyon or tachyonic particle is a hypothetical particle that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are inconsistent with the known laws of physics. If such particles did exist they could be used to send signals faster than light. According to the theory of relativity this would violate causality, leading to logical paradoxes such as the grandfather paradox. Tachyons would exhibit the unusual property of increasing in speed as their energy decreases, and would require infinite energy to slow to the speed of light. No verifiable experimental evidence for the existence of such particles has been found.

<span class="mw-page-title-main">Time travel</span> Hypothetical travel into the past or future

Time travel is the hypothetical activity of traveling into the past or future. Time travel is a widely recognized concept in philosophy and fiction, particularly science fiction. In fiction, time travel is typically achieved through the use of a hypothetical device known as a time machine. The idea of a time machine was popularized by H. G. Wells's 1895 novel The Time Machine.

A wormhole is a hypothetical structure connecting disparate points in spacetime, and is based on a special solution of the Einstein field equations.

Physical causality is a physical relationship between causes and effects. It is considered to be fundamental to all natural sciences and behavioural sciences, especially physics. Causality is also a topic studied from the perspectives of philosophy, statistics and logic. Causality means that an effect can not occur from a cause that is not in the back (past) light cone of that event. Similarly, a cause can not have an effect outside its front (future) light cone.

In physics, the principle of locality states that an object is influenced directly only by its immediate surroundings. A theory that includes the principle of locality is said to be a "local theory". This is an alternative to the concept of instantaneous, or "non-local" action at a distance. Locality evolved out of the field theories of classical physics. The idea is that for a cause at one point to have an effect at another point, something in the space between those points must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence.

The transactional interpretation of quantum mechanics (TIQM) takes the wave function of the standard quantum formalism, and its complex conjugate, to be retarded and advanced waves that form a quantum interaction as a Wheeler–Feynman handshake or transaction. It was first proposed in 1986 by John G. Cramer, who argues that it helps in developing intuition for quantum processes. He also suggests that it avoids the philosophical problems with the Copenhagen interpretation and the role of the observer, and also resolves various quantum paradoxes. TIQM formed a minor plot point in his science fiction novel Einstein's Bridge.

<span class="mw-page-title-main">Quantum Zeno effect</span> Quantum measurement phenomenon

The quantum Zeno effect is a feature of quantum-mechanical systems allowing a particle's time evolution to be slowed down by measuring it frequently enough with respect to some chosen measurement setting.

A Bell test, also known as Bell inequality test or Bell experiment, is a real-world physics experiment designed to test the theory of quantum mechanics in relation to Albert Einstein's concept of local realism. Named for John Stewart Bell, the experiments test whether or not the real world satisfies local realism, which requires the presence of some additional local variables to explain the behavior of particles like photons and electrons. The test empirically evaluates the implications of Bell's theorem. As of 2015, all Bell tests have found that the hypothesis of local hidden variables is inconsistent with the way that physical systems behave.

In the interpretation of quantum mechanics, a local hidden-variable theory is a hidden-variable theory that satisfies the principle of locality. These models attempt to account for the probabilistic features of quantum mechanics via the mechanism of underlying, but inaccessible variables, with the additional requirement that distant events be statistically independent.

A temporal paradox, time paradox, or time travel paradox, is a paradox, an apparent contradiction, or logical contradiction associated with the idea of time travel or other foreknowledge of the future. While the notion of time travel to the future complies with the current understanding of physics via relativistic time dilation, temporal paradoxes arise from circumstances involving hypothetical time travel to the past – and are often used to demonstrate its impossibility.

Quantum mechanics is the study of matter and its interactions with energy on the scale of atomic and subatomic particles. By contrast, classical physics explains matter and energy only on a scale familiar to human experience, including the behavior of astronomical bodies such as the moon. Classical physics is still used in much of modern science and technology. However, towards the end of the 19th century, scientists discovered phenomena in both the large (macro) and the small (micro) worlds that classical physics could not explain. The desire to resolve inconsistencies between observed phenomena and classical theory led to a revolution in physics, a shift in the original scientific paradigm: the development of quantum mechanics.

The Wheeler–Feynman absorber theory, named after its originators, the physicists, Richard Feynman, and John Archibald Wheeler, is a theory of electrodynamics based on a relativistic correct extension of action at a distance electron particles. The theory postulates no independent electromagnetic field. Rather, the whole theory is encapsulated by the Lorentz-invariant action of particle trajectories defined as

For classical dynamics at relativistic speeds, see relativistic mechanics.

In theoretical physics, quantum nonlocality refers to the phenomenon by which the measurement statistics of a multipartite quantum system do not allow an interpretation with local realism. Quantum nonlocality has been experimentally verified under a variety of physical assumptions. Any physical theory that aims at superseding or replacing quantum theory should account for such experiments and therefore cannot fulfill local realism; quantum nonlocality is a property of the universe that is independent of our description of nature.

Lucien Hardy is a British-Canadian theoretical physicist currently based at the Perimeter Institute for Theoretical Physics in Waterloo, Canada.

In physics, a tachyonic field, or simply tachyon, is a quantum field with an imaginary mass. Although tachyonic particles are a purely hypothetical concept that violate a number of essential physical principles, at least one field with imaginary mass, the Higgs field, is believed to exist. Under no circumstances do any excitations of tachyonic fields ever propagate faster than light—the presence or absence of a tachyonic (imaginary) mass has no effect on the maximum velocity of signals, and so unlike faster-than-light particles there is no violation of causality. Tachyonic fields play an important role in physics and are discussed in popular books.

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

References

  1. 1 2 Faye, Jan (2001-08-27). "Backward Causation". Stanford Encyclopedia of Philosophy. Retrieved 2006-12-24.
  2. Barry, Patrick (September 2006). "What's done is done…". New Scientist. 191 (2571): 36–39. doi:10.1016/s0262-4079(06)60613-1 . Retrieved 2006-12-19.
  3. Sheehan, Daniel P. (2006). Frontiers of Time: Retrocausation - Experiment and Theory; San Diego, California, 20-22 June 2006. Melville, New York: American Institute of Physics. ISBN   978-0735403611.
  4. Beauchamp, Tom L.; Rosenberg, Alexander (1981). Hume and the Problem of Causation. New York: Oxford University Press. ISBN   9780195202366.
  5. Shinya Moriyama, "Prajñākaragupta: Buddhist Epistemology as the Path to the Wisdom of Non-Duality", in Edelglass (ed) et al. The Routledge Handbook of Indian Buddhist Philosophy (Routledge Handbooks in Philosophy), pp. 528-539. Routledge (2022).
  6. 1 2 Dummett, A. E.; Flew, A. (11 July 1954). "Symposium: "Can An Effect Precede Its Cause?"". Aristotelian Society Supplementary Volume. 28 (1): 27–62. doi:10.1093/aristoteliansupp/28.1.27.
  7. Black, Max (January 1956). "Why Cannot an Effect Precede Its Cause?". Analysis. 16 (3): 49–58. doi:10.2307/3326929. JSTOR   3326929.
  8. Ellis, Brian (2002). The Philosophy of Nature: A Guide to the New Essentialism . Montréal: McGill-Queen's University Press. ISBN   9780773524743.
  9. Beebee, Helen (2006). Hume on Causation. London: Routledge. ISBN   9780415243391.
  10. 1 2 Price, Huw (1997). Time's Arrow & Archimedes' Point: New Directions for the Physics of Time (1st ed.). New York: Oxford University Press. ISBN   978-0195117981.
  11. Cramer, John G. (1980-07-15). "Generalized absorber theory and the Einstein-Podolsky-Rosen paradox". Physical Review D. 22 (2): 362–376. Bibcode:1980PhRvD..22..362C. doi:10.1103/PhysRevD.22.362. ISSN   0556-2821.
  12. Krasnikov, S. V. (15 March 1997). "Causality violation and paradoxes". Physical Review D. 55 (6): 3427–3430. Bibcode:1997PhRvD..55.3427K. doi:10.1103/PhysRevD.55.3427.
  13. Earman, John; Smeenk, Christopher; Wüthrich, Christian (7 May 2008). "Do the laws of physics forbid the operation of time machines?". Synthese. 169 (1): 91–124. doi: 10.1007/s11229-008-9338-2 . ISSN   0039-7857.
  14. Faye, Jan (1994). Logic and Causal Reasoning. Berlin: Akad.-Verl. ISBN   978-3050025995.
  15. Elitzur, A.; Dolev, S.; Kolenda, N. (2005). Quo Vadis Quantum Mechanics?. Berlin: Springer. ISBN   978-3540221883.
  16. Cramer, J. G. (April 2014), "Status of Nonlocal Quantum Communication Test" (PDF), UW CENPA Annual Report 2013-14, Article 7.1, retrieved September 21, 2016.
  17. 1 2 Frankel, Miriam (1 June 2024). de Lange, Catherine (ed.). "A loop in time". New Scientist . New York, New York and London, England: New Scientist Limited: 33. ISSN   2059-5387.
  18. Hawking, S. W. (15 July 1992). "Chronology protection conjecture". Physical Review D. 46 (2): 603–611. Bibcode:1992PhRvD..46..603H. doi:10.1103/PhysRevD.46.603. PMID   10014972.
  19. Thorne, Kip S. (1995). Black Holes and Time Warps: Einstein's Outrageous Legacy. New York: W.W. Norton. ISBN   978-0393312768.
  20. Gott, John Richard (2002). Time Travel in Einstein's Universe: The Physical Possibilities of Travel Through Time (1st ed.). Boston: Mariner Books. ISBN   978-0618257355.
  21. 1 2 Wheeler, John Archibald; Feynman, Richard Phillips (1 April 1945). "Interaction with the Absorber as the Mechanism of Radiation" (PDF). Reviews of Modern Physics. 17 (2–3): 157–181. Bibcode:1945RvMP...17..157W. doi:10.1103/RevModPhys.17.157.
  22. Feynman, Richard Phillips (15 September 1949). "The Theory of Positrons". Physical Review. 76 (6): 749–759. Bibcode:1949PhRv...76..749F. doi:10.1103/PhysRev.76.749. S2CID   120117564.
  23. Griffiths, D. J. (2008). Introduction to Elementary Particles (2nd ed.). John Wiley & Sons. pp. 61, 70–71. ISBN   978-3-527-40601-2.
  24. Feynman, Richard (1965-12-11). The Development of the Space-Time View of Quantum Electrodynamics (Speech). Nobel Lecture. Retrieved 2007-01-02.
  25. Nambu, Y. (1 February 1950). "The Use of the Proper Time in Quantum Electrodynamics I". Progress of Theoretical Physics. 5 (1): 82–94. Bibcode:1950PThPh...5...82N. doi:10.1143/ptp/5.1.82.
  26. Villata, M. (30 November 2011). "Reply to "Comment to a paper of M. Villata on antigravity"". Astrophysics and Space Science. 337 (1): 15–17. arXiv: 1109.1201 . Bibcode:2012Ap&SS.337...15V. doi:10.1007/s10509-011-0940-2. S2CID   118540070.
  27. Watanabe, Satosi (1955). "Symmetry of physical laws. Part III. Prediction and retrodiction". Reviews of Modern Physics. 27 (2): 179–186. Bibcode:1955RvMP...27..179W. doi:10.1103/RevModPhys.27.179. hdl:10945/47584. S2CID   122168419.
  28. Aharonov, Yakir & Lev Vaidman. "The Two-State Vector Formalism: An Updated Review" (PDF). Retrieved 2014-07-07.
  29. Rave, M. J. (22 October 2008). "Interpreting Quantum Interference Using a Berry's Phase-like Quantity". Foundations of Physics. 38 (12): 1073–1081. Bibcode:2008FoPh...38.1073R. doi:10.1007/s10701-008-9252-y. S2CID   121964032.
  30. Wharton, William R. (1998-10-28). "Backward Causation and the EPR Paradox". arXiv: quant-ph/9810060 .
  31. Costa de Beauregard, Olivier (1977). "Time Symmetry and the Einstein Paradox" (PDF). Il Nuovo Cimento (42B).
  32. Ellerman, David (2012-12-11). "A Common Fallacy in Quantum Mechanics: Why Delayed Choice Experiments do NOT imply Retrocausality". Archived from the original on 2013-06-15. Retrieved 2017-05-12.
  33. Rubin, Mark A. (2001). "Locality in the Everett Interpretation of Heisenberg-Picture Quantum Mechanics". Foundations of Physics Letters. 14 (2001): 301–322. arXiv: quant-ph/0103079 . Bibcode:2001quant.ph..3079R. doi:10.1023/A:1012357515678. S2CID   6916036.
  34. Feinberg, G. (25 July 1967). "Possibility of Faster-Than-Light Particles". Physical Review. 159 (5): 1089–1105. Bibcode:1967PhRv..159.1089F. doi:10.1103/PhysRev.159.1089.
  35. John Gribbin; Book Review of "An Experiment with Time", New Scientist, 27 August 1981, 548.
  36. Schmidt, Helmut (June 1978). "Can an effect precede its cause? A model of a noncausal world". Foundations of Physics. 8 (5–6): 463–480. Bibcode:1978FoPh....8..463S. doi:10.1007/BF00708576. S2CID   120918972.
  37. Schmidt, Helmut (June 1982). "Collapse of the state vector and psychokinetic effect". Foundations of Physics. 12 (6): 565–581. Bibcode:1982FoPh...12..565S. doi:10.1007/bf00731929. S2CID   120444688.
  38. Druckman, Daniel; Swets, John A. (1988). Enhancing Human Performance: Issues, Theories, and Techniques. Washington, D.C.: National Academy Press. ISBN   9780309037921.
  39. Stenger, Victor J. (1990). Physics and Psychics: The Search for a World Beyond the Senses. Buffalo, New York: Prometheus Books. ISBN   9780879755751.
  40. Shoup, Richard (2002). "Anomalies and constraints: can clairvoyance, precognition, and psychokinesis be accommodated with known physics?". Journal of Scientific Exploration. 16.
  41. Leibovici, L. (2001). "Effects of remote, retroactive intercessory prayer on outcomes in patients with bloodstream infection: randomised controlled trial". British Medical Journal. 323 (7327): 1450–1. doi:10.1136/bmj.323.7327.1450. PMC   61047 . PMID   11751349.
  42. Bishop, J. P. (18 December 2004). "Retroactive prayer: lots of history, not much mystery, and no science". BMJ. 329 (7480): 1444–1446. doi:10.1136/bmj.329.7480.1444. PMC   535973 . PMID   15604179.
  43. LeBel, Etienne P.; Peters, Kurt R. (January 2011). "Fearing the future of empirical psychology: Bem's (2011) evidence of psi as a case study of deficiencies in modal research practice" (PDF). Review of General Psychology. 15 (4): 371–379. doi:10.1037/a0025172. S2CID   51686730 . Retrieved 2 November 2017.