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125-541: The de Broglie–Bohm theory is an interpretation of quantum mechanics which postulates that, in addition to the wavefunction , an actual configuration of particles exists, even when unobserved. The evolution over time of the configuration of all particles is defined by a guiding equation . The evolution of the wave function over time is given by the Schrödinger equation . The theory is named after Louis de Broglie (1892–1987) and David Bohm (1917–1992). The theory

250-484: A value state, which indicates what is actually true about a system at a given time. The term "modal interpretation" now is used to describe a larger set of models that grew out of this approach. The Stanford Encyclopedia of Philosophy describes several versions, including proposals by Kochen , Dieks , Clifton, Dickson, and Bub . According to Michel Bitbol , Schrödinger's views on how to interpret quantum mechanics progressed through as many as four stages, ending with

375-411: A " measurement problem ", due to the fact that the particles have a definite configuration at all times. The Born rule in de Broglie–Bohm theory is not a postulate. Rather, in this theory, the link between the probability density and the wave function has the status of a theorem, a result of a separate postulate, the " quantum equilibrium hypothesis ", which is additional to the basic principles governing

500-521: A Lorentz-invariant foliation of space-time. Thus, Dürr et al. (1999) showed that it is possible to formally restore Lorentz invariance for the Bohm–Dirac theory by introducing additional structure. This approach still requires a foliation of space-time. While this is in conflict with the standard interpretation of relativity, the preferred foliation, if unobservable, does not lead to any empirical conflicts with relativity. In 2013, Dürr et al. suggested that

625-536: A general spacetime with curvature and torsion, the guiding equation for the four-velocity u i {\displaystyle u^{i}} of an elementary fermion particle is u i = e μ i ψ ¯ γ μ ψ ψ ¯ ψ , {\displaystyle u^{i}={\frac {e_{\mu }^{i}{\bar {\psi }}\gamma ^{\mu }\psi }{{\bar {\psi }}\psi }},} where

750-420: A generalized relativistic-invariant probabilistic interpretation of quantum theory, in which | ψ | 2 {\displaystyle |\psi |^{2}} is no longer a probability density in space, but a probability density in space-time. He uses this generalized probabilistic interpretation to formulate a relativistic-covariant version of de Broglie–Bohm theory without introducing

875-409: A kind of propositional logic suitable for understanding the apparent anomalies regarding quantum measurement, most notably those concerning composition of measurement operations of complementary variables. This research area and its name originated in the 1936 paper by Garrett Birkhoff and John von Neumann , who attempted to reconcile some of the apparent inconsistencies of classical Boolean logic with

1000-482: A mechanical system forms the cotangent bundle T ∗ Q {\displaystyle T^{*}Q} of the configuration manifold Q {\displaystyle Q} . This larger manifold is called the phase space of the system. In quantum mechanics , configuration space can be used (see for example the Mott problem ), but the classical mechanics extension to phase space cannot. Instead,

1125-517: A non-collapse view that in respects resembles the interpretations of Everett and van Fraassen. Because Schrödinger subscribed to a kind of post- Machian neutral monism , in which "matter" and "mind" are only different aspects or arrangements of the same common elements, treating the wavefunction as ontic and treating it as epistemic became interchangeable. Time-symmetric interpretations of quantum mechanics were first suggested by Walter Schottky in 1921. Several theories have been proposed that modify

1250-445: A particle's position and velocity is subject to the usual uncertainty principle constraint. The theory is considered to be a hidden-variable theory , and by embracing non-locality it satisfies Bell's inequality . The measurement problem is resolved, since the particles have definite positions at all times. Collapse is explained as phenomenological . The transactional interpretation of quantum mechanics (TIQM) by John G. Cramer

1375-613: A preferred foliation of space-time. His work also covers the extension of the Bohmian interpretation to a quantization of fields and strings. Roderick I. Sutherland at the University in Sydney has a Lagrangian formalism for the pilot wave and its beables . It draws on Yakir Aharonov 's retrocasual weak measurements to explain many-particle entanglement in a special relativistic way without the need for configuration space. The basic idea

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1500-528: A quantized version of the total energy of a classical system evolving under a real-valued potential function V {\displaystyle V} on R 3 {\displaystyle \mathbb {R} ^{3}} : For many particles, the equation is the same except that ψ {\displaystyle \psi } and V {\displaystyle V} are now on configuration space, R 3 N {\displaystyle \mathbb {R} ^{3N}} : This

1625-588: A quantum state is not an element of reality—instead it represents the degrees of belief an agent has about the possible outcomes of measurements. For this reason, some philosophers of science have deemed QBism a form of anti-realism . The originators of the interpretation disagree with this characterization, proposing instead that the theory more properly aligns with a kind of realism they call "participatory realism", wherein reality consists of more than can be captured by any putative third-person account of it. The consistent histories interpretation generalizes

1750-452: A rather different set of formalisms and notation are used in the analogous concept called quantum state space . The analog of a "point particle" becomes a single point in C P 1 {\displaystyle \mathbb {C} \mathbf {P} ^{1}} , the complex projective line , also known as the Bloch sphere . It is complex, because a quantum-mechanical wave function has

1875-455: A robot arm to obtain a particular end-effector location, and it is even possible to have the robot arm move while keeping the end effector stationary. Thus, a complete description of the arm, suitable for use in kinematics, requires the specification of all of the joint positions and angles, and not just some of them. The joint parameters of the robot are used as generalized coordinates to define configurations. The set of joint parameter values

2000-573: A rough guide to development of the mainstream view during the 1990s and 2000s, a "snapshot" of opinions was collected in a poll by Schlosshauer et al. at the "Quantum Physics and the Nature of Reality" conference of July 2011. The authors reference a similarly informal poll carried out by Max Tegmark at the "Fundamental Problems in Quantum Theory" conference in August 1997. The main conclusion of

2125-402: A subspace of the n {\displaystyle n} -rigid-body configuration space. Note, however, that in robotics, the term configuration space can also refer to a further-reduced subset: the set of reachable positions by a robot's end-effector . This definition, however, leads to complexities described by the holonomy : that is, there may be several different ways of arranging

2250-471: A system refers to the position of all constituent point particles of the system. The configuration space is insufficient to completely describe a mechanical system: it fails to take into account velocities. The set of velocities available to a system defines a plane tangent to the configuration manifold of the system. At a point q ∈ Q {\displaystyle q\in Q} , that tangent plane

2375-399: A true creation or destruction of particles does not take place. To extend de Broglie–Bohm theory to curved space ( Riemannian manifolds in mathematical parlance), one simply notes that all of the elements of these equations make sense, such as gradients and Laplacians . Thus, we use equations that have the same form as above. Topological and boundary conditions may apply in supplementing

2500-431: A universe governed by the de Broglie–Bohm dynamics, Born rule behavior is typical. The situation is thus analogous to the situation in classical statistical physics. A low- entropy initial condition will, with overwhelmingly high probability, evolve into a higher-entropy state: behavior consistent with the second law of thermodynamics is typical. There are anomalous initial conditions that would give rise to violations of

2625-536: A wavefunction, but also a well-defined configuration of the whole universe (i.e., the system as defined by the boundary conditions used in solving the Schrödinger equation). The de Broglie–Bohm theory works on particle positions and trajectories like classical mechanics but the dynamics are different. In classical mechanics, the accelerations of the particles are imparted directly by forces, which exist in physical three-dimensional space. In de Broglie–Bohm theory,

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2750-404: Is deterministic and explicitly nonlocal : the velocity of any one particle depends on the value of the guiding equation, which depends on the configuration of all the particles under consideration. Measurements are a particular case of quantum processes described by the theory—for which it yields the same quantum predictions as other interpretations of quantum mechanics. The theory does not have

2875-457: Is not distributed according to the Born rule (that is, a distribution "out of quantum equilibrium") and evolving under the de Broglie–Bohm dynamics is overwhelmingly likely to evolve dynamically into a state distributed as | ψ | 2 {\displaystyle |\psi |^{2}} . In the formulation of the de Broglie–Bohm theory, there is only a wavefunction for

3000-567: Is a tetrad . If the wave function propagates according to the curved Dirac equation, then the particle moves according to the Mathisson-Papapetrou equations of motion, which are an extension of the geodesic equation . This relativistic wave-particle duality follows from the conservation laws for the spin tensor and energy-momentum tensor , and also from the covariant Heisenberg picture equation of motion. Interpretations of quantum mechanics While some variation of

3125-453: Is a theory meant to explain the emergence of the classical world from the quantum world as due to a process of Darwinian natural selection induced by the environment interacting with the quantum system; where the many possible quantum states are selected against in favor of a stable pointer state . It was proposed in 2003 by Wojciech Zurek and a group of collaborators including Ollivier, Poulin, Paz and Blume-Kohout. The development of

3250-465: Is an extension of the de Broglie–Bohm interpretation of quantum mechanics , with the electromagnetic zero-point field (ZPF) playing a central role as the guiding pilot-wave . Modern approaches to SED, like those proposed by the group around late Gerhard Grössing, among others, consider wave and particle-like quantum effects as well-coordinated emergent systems. These emergent systems are the result of speculated and calculated sub-quantum interactions with

3375-415: Is an interpretation of quantum mechanics in which a universal wavefunction obeys the same deterministic, reversible laws at all times; in particular there is no (indeterministic and irreversible ) wavefunction collapse associated with measurement. The phenomena associated with measurement are claimed to be explained by decoherence , which occurs when states interact with the environment. More precisely,

3500-461: Is an interpretation of quantum mechanics inspired by the Wheeler–Feynman absorber theory . It describes the collapse of the wave function as resulting from a time-symmetric transaction between a possibility wave from the source to the receiver (the wave function) and a possibility wave from the receiver to source (the complex conjugate of the wave function). This interpretation of quantum mechanics

3625-444: Is called the configuration manifold of the system. Notice that this is a notion of "unrestricted" configuration space, i.e. in which different point particles may occupy the same position. In mathematics, in particular in topology, a notion of "restricted" configuration space is mostly used, in which the diagonals, representing "colliding" particles, are removed. The position of a single particle moving in ordinary Euclidean 3-space

3750-403: Is called the joint space . A robot's forward and inverse kinematics equations define maps between configurations and end-effector positions, or between joint space and configuration space. Robot motion planning uses this mapping to find a path in joint space that provides an achievable route in the configuration space of the end-effector. In classical mechanics , the configuration of

3875-453: Is defined at both slits, but each particle has a well-defined trajectory that passes through exactly one of the slits. The final position of the particle on the detector screen and the slit through which the particle passes is determined by the initial position of the particle. Such initial position is not knowable or controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. In Bohm's 1952 papers he used

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4000-404: Is defined by the vector q = ( x , y , z ) {\displaystyle q=(x,y,z)} , and therefore its configuration space is Q = R 3 {\displaystyle Q=\mathbb {R} ^{3}} . It is conventional to use the symbol q {\displaystyle q} for a point in configuration space; this is the convention in both

4125-418: Is denoted by T q Q {\displaystyle T_{q}Q} . Momentum vectors are linear functionals of the tangent plane, known as cotangent vectors; for a point q ∈ Q {\displaystyle q\in Q} , that cotangent plane is denoted by T q ∗ Q {\displaystyle T_{q}^{*}Q} . The set of positions and momenta of

4250-490: Is described using generalized coordinates ; thus, three of the coordinates might describe the location of the center of mass of the rigid body, while three more might be the Euler angles describing its orientation. There is no canonical choice of coordinates; one could also choose some tip or endpoint of the rigid body, instead of its center of mass; one might choose to use quaternions instead of Euler angles, and so on. However,

4375-414: Is given by the square modulus of the (normalized) conditional wavefunction ψ I ( t , ⋅ ) {\displaystyle \psi ^{\text{I}}(t,\cdot )} (in the terminology of Dürr et al. this fact is called the fundamental conditional probability formula ). Unlike the universal wavefunction, the conditional wavefunction of a subsystem does not always evolve by

4500-560: Is guaranteed to be true for all time by the guiding equation if the initial distribution of the particles satisfies | ψ | 2 {\displaystyle |\psi |^{2}} . For a given experiment, one can postulate this as being true and verify it experimentally. But, as argued by Dürr et al., one needs to argue that this distribution for subsystems is typical. The authors argue that | ψ | 2 {\displaystyle |\psi |^{2}} , by virtue of its equivariance under

4625-554: Is no definitive historical statement of what is the Copenhagen interpretation, and there were in particular fundamental disagreements between the views of Bohr and Heisenberg. For example, Heisenberg emphasized a sharp "cut" between the observer (or the instrument) and the system being observed, while Bohr offered an interpretation that is independent of a subjective observer or measurement or collapse, which relies on an "irreversible" or effectively irreversible process that imparts

4750-403: Is not an objective property of an individual system but is that information, obtained from a knowledge of how a system was prepared, which can be used for making predictions about future measurements. ... A quantum mechanical state being a summary of the observer's information about an individual physical system changes both by dynamical laws, and whenever the observer acquires new information about

4875-464: Is not uncommon among practitioners of quantum mechanics. Similarly Richard Feynman wrote many popularizations of quantum mechanics without ever publishing about interpretation issues like quantum measurement. Others, like Nico van Kampen and Willis Lamb , have openly criticized non-orthodox interpretations of quantum mechanics. Almost all authors below are professional physicists. Configuration space (physics) In classical mechanics ,

5000-408: Is often misattributed to Richard Feynman ). The Copenhagen interpretation is a collection of views about the meaning of quantum mechanics principally attributed to Niels Bohr and Werner Heisenberg . It is one of the oldest attitudes towards quantum mechanics, as features of it date to the development of quantum mechanics during 1925–1927, and it remains one of the most commonly taught. There

5125-402: Is on the side beyond the barrier, the pattern of detected particles shows interference fringes characteristic of waves arriving at the screen from two sources (the two slits); however, the interference pattern is made up of individual dots corresponding to particles that had arrived on the screen. The system seems to exhibit the behaviour of both waves (interference patterns) and particles (dots on

De Broglie–Bohm theory - Misplaced Pages Continue

5250-420: Is possible to calculate the exact state of the system at all intermediate times. The collapse of the wavefunction is therefore not a physical change to the system, just a change in our knowledge of it due to the second measurement. Similarly, they explain entanglement as not being a true physical state but just an illusion created by ignoring retrocausality. The point where two particles appear to "become entangled"

5375-510: Is reached, with observers having no special role. Thus, objective-collapse theories are realistic, indeterministic, no-hidden-variables theories. Standard quantum mechanics does not specify any mechanism of collapse; quantum mechanics would need to be extended if objective collapse is correct. The requirement for an extension means that objective-collapse theories are alternatives to quantum mechanics rather than interpretations of it. Examples include The most common interpretations are summarized in

5500-460: Is said to have six degrees of freedom . In this case, the configuration space Q = R 3 × S O ( 3 ) {\displaystyle Q=\mathbb {R} ^{3}\times \mathrm {SO} (3)} is six-dimensional, and a point q ∈ Q {\displaystyle q\in Q} is just a point in that space. The "location" of q {\displaystyle q} in that configuration space

5625-430: Is simply a point where each particle is being influenced by events that occur to the other particle in the future. Not all advocates of time-symmetric causality favour modifying the unitary dynamics of standard quantum mechanics. Thus a leading exponent of the two-state vector formalism, Lev Vaidman , states that the two-state vector formalism dovetails well with Hugh Everett 's many-worlds interpretation . As well as

5750-529: Is that this velocity field depends on the actual positions of all of the N {\displaystyle N} particles in the universe. As explained below, in most experimental situations, the influence of all of those particles can be encapsulated into an effective wavefunction for a subsystem of the universe. The one-particle Schrödinger equation governs the time evolution of a complex-valued wavefunction on R 3 {\displaystyle \mathbb {R} ^{3}} . The equation represents

5875-427: Is the same wavefunction as in conventional quantum mechanics. In Bohm's original papers, he discusses how de Broglie–Bohm theory results in the usual measurement results of quantum mechanics. The main idea is that this is true if the positions of the particles satisfy the statistical distribution given by | ψ | 2 {\displaystyle |\psi |^{2}} . And that distribution

6000-504: Is the sphere, i.e. Q = S 2 {\displaystyle Q=S^{2}} . For n disconnected, non-interacting point particles, the configuration space is R 3 n {\displaystyle \mathbb {R} ^{3n}} . In general, however, one is interested in the case where the particles interact: for example, they are specific locations in some assembly of gears, pulleys, rolling balls, etc. often constrained to move without slipping. In this case,

6125-457: Is to introduce a preferred foliation of space-time by hand, such that each hypersurface of the foliation defines a hypersurface of equal time. Initially, it had been considered impossible to set out a description of photon trajectories in the de Broglie–Bohm theory in view of the difficulties of describing bosons relativistically. In 1996, Partha Ghose presented a relativistic quantum-mechanical description of spin-0 and spin-1 bosons starting from

6250-433: Is unique in that it not only views the wave function as a real entity, but the complex conjugate of the wave function, which appears in the Born rule for calculating the expected value for an observable, as also real. In his treatise The Mathematical Foundations of Quantum Mechanics , John von Neumann deeply analyzed the so-called measurement problem . He concluded that the entire physical universe could be made subject to

6375-486: The Copenhagen interpretation is commonly presented in textbooks, many other interpretations have been developed. Despite nearly a century of debate and experiment, no consensus has been reached among physicists and philosophers of physics concerning which interpretation best "represents" reality. The definition of quantum theorists' terms, such as wave function and matrix mechanics , progressed through many stages. For instance, Erwin Schrödinger originally viewed

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6500-639: The Duffin–Kemmer–Petiau equation , setting out Bohmian trajectories for massive bosons and for massless bosons (and therefore photons ). In 2001, Jean-Pierre Vigier emphasized the importance of deriving a well-defined description of light in terms of particle trajectories in the framework of either the Bohmian mechanics or the Nelson stochastic mechanics. The same year, Ghose worked out Bohmian photon trajectories for specific cases. Subsequent weak-measurement experiments yielded trajectories that coincide with

6625-522: The Hamiltonian formulation of classical mechanics , and in Lagrangian mechanics . The symbol p {\displaystyle p} is used to denote momenta; the symbol q ˙ = d q / d t {\displaystyle {\dot {q}}=dq/dt} refers to velocities. A particle might be constrained to move on a specific manifold . For example, if

6750-411: The tangent space T Q {\displaystyle TQ} corresponds to the velocities of the points q ∈ Q {\displaystyle q\in Q} , while the cotangent space T ∗ Q {\displaystyle T^{*}Q} corresponds to momenta. (Velocities and momenta can be connected; for the most general, abstract case, this is done with

6875-545: The " Copenhagen interpretation ", though physicists and historians of physics have argued that this terminology obscures differences between the views so designated. Copenhagen-type ideas were never universally embraced, and challenges to a perceived Copenhagen orthodoxy gained increasing attention in the 1950s with the pilot-wave interpretation of David Bohm and the many-worlds interpretation of Hugh Everett III . The physicist N. David Mermin once quipped, "New interpretations appear every year. None ever disappear." As

7000-422: The 1950s antirealism has adopted a more modest approach, often in the form of instrumentalism , permitting talk of unobservables but ultimately discarding the very question of realism and positing scientific theory as a tool to help us make predictions, not to attain a deep metaphysical understanding of the world. The instrumentalist view is typified by David Mermin 's famous slogan: "Shut up and calculate" (which

7125-457: The Everett interpretation received 17% of the vote, which is similar to the number of votes (18%) in our poll." Some concepts originating from studies of interpretations have found more practical application in quantum information science . More or less, all interpretations of quantum mechanics share two qualities: Two qualities vary among interpretations: In the philosophy of science ,

7250-831: The Hamiltonian does not contain an interaction term between subsystems (I) and (II), then ψ I {\displaystyle \psi ^{\text{I}}} does satisfy a Schrödinger equation. More generally, assume that the universal wave function ψ {\displaystyle \psi } can be written in the form where ϕ {\displaystyle \phi } solves Schrödinger equation and, ϕ ( t , q I , Q II ( t ) ) = 0 {\displaystyle \phi (t,q^{\text{I}},Q^{\text{II}}(t))=0} for all t {\displaystyle t} and q I {\displaystyle q^{\text{I}}} . Then, again,

7375-626: The Schrödinger equation (the universal wave function). He also described how measurement could cause a collapse of the wave function. This point of view was prominently expanded on by Eugene Wigner , who argued that human experimenter consciousness (or maybe even dog consciousness) was critical for the collapse, but he later abandoned this interpretation. However, consciousness remains a mystery. The origin and place in nature of consciousness are not well understood. Some specific proposals for consciousness caused wave-function collapse have been shown to be unfalsifiable. Quantum logic can be regarded as

7500-402: The Schrödinger equation, but in many situations it does. For instance, if the universal wavefunction factors as then the conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to ψ I {\displaystyle \psi ^{\text{I}}} (this is what standard quantum theory would regard as the wavefunction of subsystem (I)). If, in addition,

7625-535: The act of "observing" or "measuring"; the theory avoids assuming definite values from unperformed experiments . Copenhagen-type interpretations hold that quantum descriptions are objective, in that they are independent of physicists' mental arbitrariness. The statistical interpretation of wavefunctions due to Max Born differs sharply from Schrödinger's original intent, which was to have a theory with continuous time evolution and in which wavefunctions directly described physical reality. The many-worlds interpretation

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7750-433: The actual configuration of subsystem (I) and of the rest of the universe. For simplicity, we consider here only the spinless case. The conditional wavefunction of subsystem (I) is defined by It follows immediately from the fact that Q ( t ) = ( Q I ( t ) , Q II ( t ) ) {\displaystyle Q(t)=(Q^{\text{I}}(t),Q^{\text{II}}(t))} satisfies

7875-426: The authors is that "the Copenhagen interpretation still reigns supreme", receiving the most votes in their poll (42%), besides the rise to mainstream notability of the many-worlds interpretations: "The Copenhagen interpretation still reigns supreme here, especially if we lump it together with intellectual offsprings such as information-based interpretations and the quantum Bayesian interpretation. In Tegmark's poll,

8000-475: The classical behavior of "observation" or "measurement". Features common to Copenhagen-type interpretations include the idea that quantum mechanics is intrinsically indeterministic, with probabilities calculated using the Born rule , and the principle of complementarity , which states certain pairs of complementary properties cannot all be observed or measured simultaneously. Moreover, properties only result from

8125-519: The complete description of the individual systems leads to unnatural theoretical interpretations, which become immediately unnecessary if one accepts the interpretation that the description refers to ensembles of systems and not to individual systems. The most prominent current advocate of the ensemble interpretation is Leslie E. Ballentine, professor at Simon Fraser University , author of the text book Quantum Mechanics, A Modern Development . The de Broglie–Bohm theory of quantum mechanics (also known as

8250-424: The complex-valued wavefunction on configuration space. For a spinless single particle moving in R 3 {\displaystyle \mathbb {R} ^{3}} , the particle's velocity is For many particles labeled Q k {\displaystyle \mathbf {Q} _{k}} for the k {\displaystyle k} -th particle their velocities are The main fact to notice

8375-506: The conditional wavefunction of a subsystem does not always evolve by the Schrödinger equation is related to the fact that the usual collapse rule of standard quantum theory emerges from the Bohmian formalism when one considers conditional wavefunctions of subsystems. Pilot-wave theory is explicitly nonlocal, which is in ostensible conflict with special relativity . Various extensions of "Bohm-like" mechanics exist that attempt to resolve this problem. Bohm himself in 1953 presented an extension of

8500-406: The conditional wavefunction of subsystem (I) is (up to an irrelevant scalar factor) equal to ψ I {\displaystyle \psi ^{\text{I}}} , and if the Hamiltonian does not contain an interaction term between subsystems (I) and (II), then ψ I {\displaystyle \psi ^{\text{I}}} satisfies a Schrödinger equation. The fact that

8625-649: The configuration space is not all of R 3 n {\displaystyle \mathbb {R} ^{3n}} , but the subspace (submanifold) of allowable positions that the points can take. The set of coordinates that define the position of a reference point and the orientation of a coordinate frame attached to a rigid body in three-dimensional space form its configuration space, often denoted R 3 × S O ( 3 ) {\displaystyle \mathbb {R} ^{3}\times \mathrm {SO} (3)} where R 3 {\displaystyle \mathbb {R} ^{3}} represents

8750-397: The configuration variables associated to some subsystem (I) of the universe, and q II {\displaystyle q^{\text{II}}} denotes the remaining configuration variables. Denote respectively by Q I ( t ) {\displaystyle Q^{\text{I}}(t)} and Q II ( t ) {\displaystyle Q^{\text{II}}(t)}

8875-409: The conventional Copenhagen interpretation and attempts to provide a natural interpretation of quantum cosmology . The theory is based on a consistency criterion that allows the history of a system to be described so that the probabilities for each history obey the additive rules of classical probability. It is claimed to be consistent with the Schrödinger equation . According to this interpretation,

9000-515: The coordinates of the origin of the frame attached to the body, and S O ( 3 ) {\displaystyle \mathrm {SO} (3)} represents the rotation matrices that define the orientation of this frame relative to a ground frame. A configuration of the rigid body is defined by six parameters, three from R 3 {\displaystyle \mathbb {R} ^{3}} and three from S O ( 3 ) {\displaystyle \mathrm {SO} (3)} , and

9125-544: The description. In what follows below, the setup for one particle moving in R 3 {\displaystyle \mathbb {R} ^{3}} is given followed by the setup for N particles moving in 3 dimensions. In the first instance, configuration space and real space are the same, while in the second, real space is still R 3 {\displaystyle \mathbb {R} ^{3}} , but configuration space becomes R 3 N {\displaystyle \mathbb {R} ^{3N}} . While

9250-523: The distinction between knowledge and reality is termed epistemic versus ontic . A general law can be seen as a generalisation of the regularity of outcomes (epistemic), whereas a causal mechanism may be thought of as determining or regulating outcomes (ontic). A phenomenon can be interpreted either as ontic or as epistemic. For instance, indeterminism may be attributed to limitations of human observation and perception (epistemic), or may be explained as intrinsic physical randomness (ontic). Confusing

9375-415: The dynamical evolution of the system, is the appropriate measure of typicality for initial conditions of the positions of the particles. The authors then prove that the vast majority of possible initial configurations will give rise to statistics obeying the Born rule (i.e., | ψ | 2 {\displaystyle |\psi |^{2}} ) for measurement outcomes. In summary, in

9500-461: The electron's wave function as its charge density smeared across space, but Max Born reinterpreted the absolute square value of the wave function as the electron's probability density distributed across space; the Born rule , as it is now called, matched experiment, whereas Schrödinger's charge density view did not. The views of several early pioneers of quantum mechanics, such as Niels Bohr and Werner Heisenberg , are often grouped together as

9625-609: The entire universe (which always evolves by the Schrödinger equation). Here, the "universe" is simply the system limited by the same boundary conditions used to solve the Schrödinger equation. However, once the theory is formulated, it is convenient to introduce a notion of wavefunction also for subsystems of the universe. Let us write the wavefunction of the universe as ψ ( t , q I , q II ) {\displaystyle \psi (t,q^{\text{I}},q^{\text{II}})} , where q I {\displaystyle q^{\text{I}}} denotes

9750-432: The epistemic with the ontic—if for example one were to presume that a general law actually "governs" outcomes, and that the statement of a regularity has the role of a causal mechanism—is a category mistake . In a broad sense, scientific theory can be viewed as offering an approximately true description or explanation of the natural world ( scientific realism ) or as providing nothing more than an account of our knowledge of

9875-472: The equations of quantum mechanics to be symmetric with respect to time reversal. (See Wheeler–Feynman time-symmetric theory .) This creates retrocausality : events in the future can affect ones in the past, exactly as events in the past can affect ones in the future. In these theories, a single measurement cannot fully determine the state of a system (making them a type of hidden-variables theory ), but given two measurements performed at different times, it

10000-415: The evolution of Schrödinger's equation. For a de Broglie–Bohm theory on curved space with spin, the spin space becomes a vector bundle over configuration space, and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space. The field equations for the de Broglie–Bohm theory in the relativistic case with spin can also be given for curved space-times with torsion. In

10125-478: The fact that Q ( t ) {\displaystyle Q(t)} is random with probability density given by the square modulus of ψ ( t , ⋅ ) {\displaystyle \psi (t,\cdot )} implies that the conditional probability density of Q I ( t ) {\displaystyle Q^{\text{I}}(t)} given Q II ( t ) {\displaystyle Q^{\text{II}}(t)}

10250-418: The facts related to measurement and observation in quantum mechanics. Modal interpretations of quantum mechanics were first conceived of in 1972 by Bas van Fraassen , in his paper "A formal approach to the philosophy of science". Van Fraassen introduced a distinction between a dynamical state, which describes what might be true about a system and which always evolves according to the Schrödinger equation, and

10375-456: The fields of quantum information and Bayesian probability and aims to eliminate the interpretational conundrums that have beset quantum theory. QBism deals with common questions in the interpretation of quantum theory about the nature of wavefunction superposition , quantum measurement , and entanglement . According to QBism, many, but not all, aspects of the quantum formalism are subjective in nature. For example, in this interpretation,

10500-399: The fullest extent. The interpretation states that the wave function does not apply to an individual system – for example, a single particle – but is an abstract statistical quantity that only applies to an ensemble (a vast multitude) of similarly prepared systems or particles. In the words of Einstein: The attempt to conceive the quantum-theoretical description as

10625-452: The guiding equation that also the configuration Q I ( t ) {\displaystyle Q^{\text{I}}(t)} satisfies a guiding equation identical to the one presented in the formulation of the theory, with the universal wavefunction ψ {\displaystyle \psi } replaced with the conditional wavefunction ψ I {\displaystyle \psi ^{\text{I}}} . Also,

10750-554: The guiding equation with a fixed number of particles. But under a stochastic process , particles may be created and annihilated. The distribution of creation events is dictated by the wavefunction. The wavefunction itself is evolving at all times over the full multi-particle configuration space. Hrvoje Nikolić introduces a purely deterministic de Broglie–Bohm theory of particle creation and destruction, according to which particle trajectories are continuous, but particle detectors behave as if particles have been created or destroyed even when

10875-408: The interference is constructive, resulting in the interference pattern on the detector screen. To explain the behavior when the particle is detected to go through one slit, one needs to appreciate the role of the conditional wavefunction and how it results in the collapse of the wavefunction; this is explained below. The basic idea is that the environment registering the detection effectively separates

11000-435: The mainstream interpretations discussed above, a number of other interpretations have been proposed that have not made a significant scientific impact for whatever reason. These range from proposals by mainstream physicists to the more occult ideas of quantum mysticism . Some ideas are discussed in the context of interpreting quantum mechanics but are not necessarily regarded as interpretations themselves. Quantum Darwinism

11125-403: The natural world ( antirealism ). A realist stance sees the epistemic as giving us a window onto the ontic, whereas an antirealist stance sees the epistemic as providing only a logically consistent picture of the ontic. In the first half of the 20th Century, a key antirealist philosophy was logical positivism , which sought to exclude unobservable aspects of reality from scientific theory. Since

11250-404: The no-signal theorems of quantum theory. Just as special relativity is a limiting case of general relativity when the spacetime curvature vanishes, so, too is statistical no-entanglement signaling quantum theory with the Born rule a limiting case of the post-quantum action-reaction Lagrangian when the reaction is set to zero and the final boundary condition is integrated out. To incorporate spin ,

11375-415: The observer, with respect to the observed system. However, it is held by relational quantum mechanics that this applies to all physical objects, whether or not they are conscious or macroscopic. Any "measurement event" is seen simply as an ordinary physical interaction, an establishment of the sort of correlation discussed above. Thus the physical content of the theory has to do not with objects themselves, but

11500-426: The parameterization does not change the mechanical characteristics of the system; all of the different parameterizations ultimately describe the same (six-dimensional) manifold, the same set of possible positions and orientations. Some parameterizations are easier to work with than others, and many important statements can be made by working in a coordinate-free fashion. Examples of coordinate-free statements are that

11625-407: The parameters that define the configuration of a system are called generalized coordinates , and the space defined by these coordinates is called the configuration space of the physical system . It is often the case that these parameters satisfy mathematical constraints, such that the set of actual configurations of the system is a manifold in the space of generalized coordinates. This manifold

11750-436: The particle is attached to a rigid linkage, free to swing about the origin, it is effectively constrained to lie on a sphere. Its configuration space is the subset of coordinates in R 3 {\displaystyle \mathbb {R} ^{3}} that define points on the sphere S 2 {\displaystyle S^{2}} . In this case, one says that the manifold Q {\displaystyle Q}

11875-454: The particle positions themselves are in real space, the velocity field and wavefunction are on configuration space, which is how particles are entangled with each other in this theory. Extensions to this theory include spin and more complicated configuration spaces. We use variations of Q {\displaystyle \mathbf {Q} } for particle positions, while ψ {\displaystyle \psi } represents

12000-497: The parts of the wavefunction describing observers become increasingly entangled with the parts of the wavefunction describing their experiments. Although all possible outcomes of experiments continue to lie in the wavefunction's support, the times at which they become correlated with observers effectively "split" the universe into mutually unobservable alternate histories . Quantum informational approaches have attracted growing support. They subdivide into two kinds. The state

12125-461: The physical theory stands, and is consistent with itself and with reality; difficulties arise only when one attempts to "interpret" the theory. Nevertheless, designing experiments that would test the various interpretations is the subject of active research. Most of these interpretations have variants. For example, it is difficult to get a precise definition of the Copenhagen interpretation as it

12250-415: The pilot wave theory) is a theory by Louis de Broglie and extended later by David Bohm to include measurements. Particles, which always have positions, are guided by the wavefunction. The wavefunction evolves according to the Schrödinger wave equation , and the wavefunction never collapses. The theory takes place in a single spacetime, is non-local , and is deterministic. The simultaneous determination of

12375-400: The position of the particle. The wavefunction itself, and not the particles, determines the dynamical evolution of the system: the particles do not act back onto the wave function. As Bohm and Hiley worded it, "the Schrödinger equation for the quantum field does not have sources, nor does it have any other way by which the field could be directly affected by the condition of the particles [...]

12500-429: The predicted trajectories. The significance of these experimental findings is controversial. Chris Dewdney and G. Horton have proposed a relativistically covariant, wave-functional formulation of Bohm's quantum field theory and have extended it to a form that allows the inclusion of gravity. Nikolić has proposed a Lorentz-covariant formulation of the Bohmian interpretation of many-particle wavefunctions. He has developed

12625-412: The purpose of a quantum-mechanical theory is to predict the relative probabilities of various alternative histories (for example, of a particle). The ensemble interpretation , also called the statistical interpretation, can be viewed as a minimalist interpretation. That is, it claims to make the fewest assumptions associated with the standard mathematics. It takes the statistical interpretation of Born to

12750-417: The quantum "field exerts a new kind of "quantum-mechanical" force". Bohm hypothesized that each particle has a "complex and subtle inner structure" that provides the capacity to react to the information provided by the wavefunction by the quantum potential. Also, unlike in classical mechanics, physical properties (e.g., mass, charge) are spread out over the wavefunction in de Broglie–Bohm theory, not localized at

12875-420: The quantum theory can be understood completely in terms of the assumption that the quantum field has no sources or other forms of dependence on the particles". P. Holland considers this lack of reciprocal action of particles and wave function to be one "[a]mong the many nonclassical properties exhibited by this theory". Holland later called this a merely apparent lack of back reaction, due to the incompleteness of

13000-431: The rather abstract notion of the tautological one-form .) For a robotic arm consisting of numerous rigid linkages, the configuration space consists of the location of each linkage (taken to be a rigid body, as in the section above), subject to the constraints of how the linkages are attached to each other, and their allowed range of motion. Thus, for n {\displaystyle n} linkages, one might consider

13125-434: The relations between them. QBism , which originally stood for "quantum Bayesianism", is an interpretation of quantum mechanics that takes an agent's actions and experiences as the central concerns of the theory. This interpretation is distinguished by its use of a subjective Bayesian account of probabilities to understand the quantum mechanical Born rule as a normative addition to good decision-making. QBism draws from

13250-636: The required foliation could be covariantly determined by the wavefunction. The relation between nonlocality and preferred foliation can be better understood as follows. In de Broglie–Bohm theory, nonlocality manifests as the fact that the velocity and acceleration of one particle depends on the instantaneous positions of all other particles. On the other hand, in the theory of relativity the concept of instantaneousness does not have an invariant meaning. Thus, to define particle trajectories, one needs an additional rule that defines which space-time points should be considered instantaneous. The simplest way to achieve this

13375-438: The same time, it may be in a superposition of two or more states. Consequently, if quantum mechanics is to be a complete theory, relational quantum mechanics argues that the notion of "state" describes not the observed system itself, but the relationship, or correlation, between the system and its observer(s). The state vector of conventional quantum mechanics becomes a description of the correlation of some degrees of freedom in

13500-399: The screen). If this experiment is modified so that one slit is closed, no interference pattern is observed. Thus, the state of both slits affects the final results. It can also be arranged to have a minimally invasive detector at one of the slits to detect which slit the particle went through. When that is done, the interference pattern disappears. In de Broglie–Bohm theory, the wavefunction

13625-443: The second law; however in the absence of some very detailed evidence supporting the realization of one of those conditions, it would be quite unreasonable to expect anything but the actually observed uniform increase of entropy. Similarly in the de Broglie–Bohm theory, there are anomalous initial conditions that would produce measurement statistics in violation of the Born rule (conflicting the predictions of standard quantum theory), but

13750-429: The state is a construct of the observer and not an objective property of the physical system. The essential idea behind relational quantum mechanics , following the precedent of special relativity , is that different observers may give different accounts of the same series of events: for example, to one observer at a given point in time, a system may be in a single, "collapsed" eigenstate , while to another observer at

13875-407: The system through the process of measurement. The existence of two laws for the evolution of the state vector ... becomes problematical only if it is believed that the state vector is an objective property of the system ... The "reduction of the wavepacket" does take place in the consciousness of the observer, not because of any unique physical process which takes place there, but only because

14000-435: The table below. The values shown in the cells of the table are not without controversy, for the precise meanings of some of the concepts involved are unclear and, in fact, are themselves at the center of the controversy surrounding the given interpretation. For another table comparing interpretations of quantum theory, see reference. No experimental evidence exists that distinguishes among these interpretations. To that extent,

14125-524: The theory is due to the integration of a number of Zurek's research topics pursued over the course of twenty-five years including pointer states , einselection and decoherence . Objective-collapse theories differ from the Copenhagen interpretation by regarding both the wave function and the process of collapse as ontologically objective (meaning these exist and occur independent of the observer). In objective theories, collapse occurs either randomly ("spontaneous localization") or when some physical threshold

14250-456: The theory satisfying the Dirac equation for a single particle. However, this was not extensible to the many-particle case because it used an absolute time. A renewed interest in constructing Lorentz-invariant extensions of Bohmian theory arose in the 1990s; see Bohm and Hiley: The Undivided Universe and references therein. Another approach is given by Dürr et al., who use Bohm–Dirac models and

14375-410: The total space [ R 3 × S O ( 3 ) ] n {\displaystyle \left[\mathbb {R} ^{3}\times \mathrm {SO} (3)\right]^{n}} except that all of the various attachments and constraints mean that not every point in this space is reachable. Thus, the configuration space Q {\displaystyle Q} is necessarily

14500-525: The two wave packets in configuration space. The de Broglie–Bohm theory describes a pilot wave ψ ( q , t ) ∈ C {\displaystyle \psi (q,t)\in \mathbb {C} } in a configuration space Q {\displaystyle Q} and trajectories q ( t ) ∈ Q {\displaystyle q(t)\in Q} of particles as in classical mechanics but defined by non-Newtonian mechanics. At every moment of time there exists not only

14625-416: The typicality theorem shows that absent some specific reason to believe one of those special initial conditions was in fact realized, the Born rule behavior is what one should expect. It is in this qualified sense that the Born rule is, for the de Broglie–Bohm theory, a theorem rather than (as in ordinary quantum theory) an additional postulate . It can also be shown that a distribution of particles which

14750-411: The wave function ψ {\displaystyle \psi } is a spinor , ψ ¯ {\displaystyle {\bar {\psi }}} is the corresponding adjoint , γ μ {\displaystyle \gamma ^{\mu }} are the Dirac matrices , and e μ i {\displaystyle e_{\mu }^{i}}

14875-408: The wave function. There are several equivalent mathematical formulations of the theory. De Broglie–Bohm theory is based on the following postulates: Even though this latter relation is frequently presented as an axiom of the theory, Bohm presented it as derivable from statistical-mechanical arguments in the original papers of 1952. This argument was further supported by the work of Bohm in 1953 and

15000-1805: The wavefunction becomes complex-vector-valued. The value space is called spin space; for a spin-1/2 particle, spin space can be taken to be C 2 {\displaystyle \mathbb {C} ^{2}} . The guiding equation is modified by taking inner products in spin space to reduce the complex vectors to complex numbers. The Schrödinger equation is modified by adding a Pauli spin term : d Q k d t ( t ) = ℏ m k Im ⁡ ( ( ψ , D k ψ ) ( ψ , ψ ) ) ( Q 1 , … , Q N , t ) , i ℏ ∂ ∂ t ψ = ( − ∑ k = 1 N ℏ 2 2 m k D k 2 + V − ∑ k = 1 N μ k S k ℏ s k ⋅ B ( q k ) ) ψ , {\displaystyle {\begin{aligned}{\frac {d\mathbf {Q} _{k}}{dt}}(t)&={\frac {\hbar }{m_{k}}}\operatorname {Im} \left({\frac {(\psi ,D_{k}\psi )}{(\psi ,\psi )}}\right)(\mathbf {Q} _{1},\ldots ,\mathbf {Q} _{N},t),\\i\hbar {\frac {\partial }{\partial t}}\psi &=\left(-\sum _{k=1}^{N}{\frac {\hbar ^{2}}{2m_{k}}}D_{k}^{2}+V-\sum _{k=1}^{N}\mu _{k}{\frac {\mathbf {S} _{k}}{\hbar s_{k}}}\cdot \mathbf {B} (\mathbf {q} _{k})\right)\psi ,\end{aligned}}} where Stochastic electrodynamics (SED)

15125-453: The wavefunction to construct a quantum potential that, when included in Newton's equations, gave the trajectories of the particles streaming through the two slits. In effect the wavefunction interferes with itself and guides the particles by the quantum potential in such a way that the particles avoid the regions in which the interference is destructive and are attracted to the regions in which

15250-406: The zero-point field. In Dürr et al., the authors describe an extension of de Broglie–Bohm theory for handling creation and annihilation operators , which they refer to as "Bell-type quantum field theories". The basic idea is that configuration space becomes the (disjoint) space of all possible configurations of any number of particles. For part of the time, the system evolves deterministically under

15375-412: Was already published by Costa de Beauregard in the 1950s and is also used by John Cramer in his transactional interpretation except the beables that exist between the von Neumann strong projection operator measurements. Sutherland's Lagrangian includes two-way action-reaction between pilot wave and beables. Therefore, it is a post-quantum non-statistical theory with final boundary conditions that violate

15500-400: Was developed and argued by many people. Although interpretational opinions are openly and widely discussed today, that was not always the case. A notable exponent of a tendency of silence was Paul Dirac who once wrote: "The interpretation of quantum mechanics has been dealt with by many authors, and I do not want to discuss it here. I want to deal with more fundamental things." This position

15625-413: Was substantiated by Vigier and Bohm's paper of 1954, in which they introduced stochastic fluid fluctuations that drive a process of asymptotic relaxation from quantum non-equilibrium to quantum equilibrium (ρ → |ψ|). The double-slit experiment is an illustration of wave–particle duality . In it, a beam of particles (such as electrons) travels through a barrier that has two slits. If a detector screen

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