Position and momentum spaces

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Position_and_momentum_spaces

In physics and geometry, there are two closely related vector spaces, usually three-dimensional but in general of any finite dimension. Position space (also real space or coordinate space) is the set of all position vectors r in Euclidean space, and has dimensions of length; a position vector defines a point in space. (If the position vector of a point particle varies with time, it will trace out a path, the trajectory of a particle.) Momentum space is the set of all momentum vectors p a physical system can have; the momentum vector of a particle corresponds to its motion, with dimension of mass ⋅ length ⋅ time⁻¹.

Mathematically, the duality between position and momentum is an example of Pontryagin duality. In particular, if a function is given in position space, f(r), then its Fourier transform obtains the function in momentum space, φ(p). Conversely, the inverse Fourier transform of a momentum space function is a position space function.

These quantities and ideas transcend all of classical and quantum physics, and a physical system can be described using either the positions of the constituent particles, or their momenta, both formulations equivalently provide the same information about the system in consideration. Another quantity is useful to define in the context of waves. The wave vector k (or simply "k-vector") has dimensions of reciprocal length, making it an analogue of angular frequency ω which has dimensions of reciprocal time. The set of all wave vectors is k-space. Usually, the position vector r is more intuitive and simpler than the wave vector k, though the converse can also be true, such as in solid-state physics.

Quantum mechanics provides two fundamental examples of the duality between position and momentum, the Heisenberg uncertainty principle ΔxΔp ≥ ħ/2 stating that position and momentum cannot be simultaneously known to arbitrary precision, and the de Broglie relation p = ħk which states the momentum and wavevector of a free particle are proportional to each other. In this context, when it is unambiguous, the terms "momentum" and "wavevector" are used interchangeably. However, the de Broglie relation is not true in a crystal.

Classical mechanics

Lagrangian mechanics

Most often in Lagrangian mechanics, the Lagrangian L(q, dq/dt, t) is in configuration space, where q = (q₁, q₂,..., q_n) is an n-tuple of the generalized coordinates. The Euler–Lagrange equations of motion are $${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i}=\frac{\mathrm{\partial }L}{\mathrm{\partial }{q}_i},{\dot{q}}_i\equiv \frac{d{q}_i}{dt}.}$$

(One overdot indicates one time derivative). Introducing the definition of canonical momentum for each generalized coordinate $${\displaystyle p_i=\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i},}$$ the Euler–Lagrange equations take the form $${\displaystyle {\dot{p}}_i=\frac{\mathrm{\partial }L}{\mathrm{\partial }{q}_i}.}$$

The Lagrangian can be expressed in momentum space also, L′(p, dp/dt, t), where p = (p₁, p₂, ..., p_n) is an n-tuple of the generalized momenta. A Legendre transformation is performed to change the variables in the total differential of the generalized coordinate space Lagrangian; $${\displaystyle dL=\sum _{i=1}^n\left(\frac{\mathrm{\partial }L}{\mathrm{\partial }{q}_i}d{q}_i+\frac{\mathrm{\partial }L}{\mathrm{\partial }{\dot{q}}_i}d{\dot{q}}_i\right)+\frac{\mathrm{\partial }L}{\mathrm{\partial }t}dt=\sum _{i=1}^n({\dot{p}}_{i}d{q}_i+p_{i}d{\dot{q}}_i)+\frac{\mathrm{\partial }L}{\mathrm{\partial }t}dt,}$$ where the definition of generalized momentum and Euler–Lagrange equations have replaced the partial derivatives of L. The product rule for differentials allows the exchange of differentials in the generalized coordinates and velocities for the differentials in generalized momenta and their time derivatives, $${\displaystyle {\dot{p}}_{i}d{q}_i=d(q_i{\dot{p}}_i)-q_{i}d{\dot{p}}_i}$$ $${\displaystyle p_{i}d{\dot{q}}_i=d({\dot{q}}_i{p}_i)-{\dot{q}}_{i}d{p}_i}$$ which after substitution simplifies and rearranges to $${\displaystyle d\left[L-\sum _{i=1}^n(q_i{\dot{p}}_i+{\dot{q}}_i{p}_i)\right]=-\sum _{i=1}^n({\dot{q}}_{i}d{p}_i+q_{i}d{\dot{p}}_i)+\frac{\mathrm{\partial }L}{\mathrm{\partial }t}dt.}$$

Now, the total differential of the momentum space Lagrangian L′ is $${\displaystyle d{L}^{\prime }=\sum _{i=1}^n\left(\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{p}_i}d{p}_i+\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{\dot{p}}_i}d{\dot{p}}_i\right)+\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }t}dt}$$ so by comparison of differentials of the Lagrangians, the momenta, and their time derivatives, the momentum space Lagrangian L′ and the generalized coordinates derived from L′ are respectively $${\displaystyle L^{\prime }=L-\sum _{i=1}^n(q_i{\dot{p}}_i+{\dot{q}}_i{p}_i),-{\dot{q}}_i=\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{p}_i},-q_i=\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{\dot{p}}_i}.}$$

Combining the last two equations gives the momentum space Euler–Lagrange equations $${\displaystyle \frac{d}{dt}\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{\dot{p}}_i}=\frac{\mathrm{\partial }{L}^{\prime }}{\mathrm{\partial }{p}_i}.}$$

The advantage of the Legendre transformation is that the relation between the new and old functions and their variables are obtained in the process. Both the coordinate and momentum forms of the equation are equivalent and contain the same information about the dynamics of the system. This form may be more useful when momentum or angular momentum enters the Lagrangian.

Hamiltonian mechanics

In Hamiltonian mechanics, unlike Lagrangian mechanics which uses either all the coordinates or the momenta, the Hamiltonian equations of motion place coordinates and momenta on equal footing. For a system with Hamiltonian H(q, p, t), the equations are $${\displaystyle {\dot{q}}_i=\frac{\mathrm{\partial }H}{\mathrm{\partial }{p}_i},{\dot{p}}_i=-\frac{\mathrm{\partial }H}{\mathrm{\partial }{q}_i}.}$$

Quantum mechanics

In quantum mechanics, a particle is described by a quantum state. This quantum state can be represented as a superposition of basis states. In principle one is free to choose the set of basis states, as long as they span the state space. If one chooses the (generalized) eigenfunctions of the position operator as a set of basis functions, one speaks of a state as a wave function ψ(r) in position space. The familiar Schrödinger equation in terms of the position r is an example of quantum mechanics in the position representation.

By choosing the eigenfunctions of a different operator as a set of basis functions, one can arrive at a number of different representations of the same state. If one picks the eigenfunctions of the momentum operator as a set of basis functions, the resulting wave function ${\displaystyle \varphi (\mathbf{k})}$ is said to be the wave function in momentum space.

A feature of quantum mechanics is that phase spaces can come in different types: discrete-variable, rotor, and continuous-variable. The table below summarizes some relations involved in the three types of phase spaces.

Comparison and summary of relations between conjugate variables in discrete-variable (DV), rotor (ROT), and continuous-variable (CV) phase spaces (taken from arXiv:1709.04460). Most physically relevant phase spaces consist of combinations of these three. Each phase space consists of position and momentum, whose possible values are taken from a locally compact Abelian group and its dual. A quantum mechanical state can be fully represented in terms of either variables, and the transformation used to go between position and momentum spaces is, in each of the three cases, a variant of the Fourier transform. The table uses bra-ket notation as well as mathematical terminology describing Canonical commutation relations (CCR).

Reciprocal relation

See also: Wave packet

The momentum representation of a wave function and the de Broglie relation are closely related to the Fourier inversion theorem and the concept of frequency domain. Since a free particle has a spatial frequency ${\displaystyle k=|\mathbf{k}|=2\pi /\lambda }$ proportional to the momentum ${\displaystyle p=|\mathbf{p}|=\hslash k}$, describing the particle as a sum of frequency components is equivalent to describing it as the Fourier transform of a "sufficiently nice" wave function in momentum space.

Position space

Further information: Position operator

Suppose we have a three-dimensional wave function in position space ψ(r), then we can write this functions as a weighted sum of orthogonal basis functions ψ_j(r): $${\displaystyle \psi (\mathbf{r})=\sum _j{\varphi }_j{\psi }_j(\mathbf{r})}$$ or, in the continuous case, as an integral $${\displaystyle \psi (\mathbf{r})={\int }_{\mathbf{k}\text{-space}}\varphi (\mathbf{k}){\psi }_{\mathbf{k}}(\mathbf{r}){\mathrm{d}}^3\mathbf{k}}$$ It is clear that if we specify the set of functions ${\displaystyle {\psi }_{\mathbf{k}}(\mathbf{r})}$, say as the set of eigenfunctions of the momentum operator, the function ${\displaystyle \varphi (\mathbf{k})}$ holds all the information necessary to reconstruct ψ(r) and is therefore an alternative description for the state ${\displaystyle \psi }$.

In coordinate representation the momentum operator is given by $${\displaystyle \hat{\mathbf{p}}=-i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{r}}}$$ (see matrix calculus for the denominator notation) with appropriate domain. The eigenfunctions are $${\displaystyle {\psi }_{\mathbf{k}}(\mathbf{r})=\frac{1}{(\sqrt{2\pi }{)}^3}{e}^{i\mathbf{k}\cdot \mathbf{r}}}$$ and eigenvalues ħk. So $${\displaystyle \psi (\mathbf{r})=\frac{1}{(\sqrt{2\pi }{)}^3}{\int }_{\mathbf{k}\text{-space}}\varphi (\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{r}}{\mathrm{d}}^3\mathbf{k}}$$ and we see that the momentum representation is related to the position representation by a Fourier transform.

Momentum space

Further information: Momentum operator

Conversely, a three-dimensional wave function in momentum space ${\displaystyle \varphi (\mathbf{k})}$ can be expressed as a weighted sum of orthogonal basis functions ${\displaystyle {\varphi }_j(\mathbf{k})}$, $${\displaystyle \varphi (\mathbf{k})=\sum _j{\psi }_j{\varphi }_j(\mathbf{k}),}$$ or as an integral, $${\displaystyle \varphi (\mathbf{k})={\int }_{\mathbf{r}\text{-space}}\psi (\mathbf{r}){\varphi }_{\mathbf{r}}(\mathbf{k}){\mathrm{d}}^3\mathbf{r}.}$$

In momentum representation the position operator is given by $${\displaystyle \hat{\mathbf{r}}=i\hslash \frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{p}}=i\frac{\mathrm{\partial }}{\mathrm{\partial }\mathbf{k}}}$$ with eigenfunctions $${\displaystyle {\varphi }_{\mathbf{r}}(\mathbf{k})=\frac{1}{{\left(\sqrt{2\pi }\right)}^3}{e}^{-i\mathbf{k}\cdot \mathbf{r}}}$$ and eigenvalues r. So a similar decomposition of ${\displaystyle \varphi (\mathbf{k})}$ can be made in terms of the eigenfunctions of this operator, which turns out to be the inverse Fourier transform, $${\displaystyle \varphi (\mathbf{k})=\frac{1}{(\sqrt{2\pi }{)}^3}{\int }_{\mathbf{r}\text{-space}}\psi (\mathbf{r})e^{-i\mathbf{k}\cdot \mathbf{r}}{\mathrm{d}}^3\mathbf{r}.}$$

Unitary equivalence

The position and momentum operators are unitarily equivalent, with the unitary operator being given explicitly by the Fourier transform, namely a quarter-cycle rotation in phase space, generated by the oscillator Hamiltonian. Thus, they have the same spectrum. In physical language, p acting on momentum space wave functions is the same as r acting on position space wave functions (under the image of the Fourier transform).

Reciprocal space and crystals

Main article: Reciprocal lattice

For an electron (or other particle) in a crystal, its value of k relates almost always to its crystal momentum, not its normal momentum. Therefore, k and p are not simply proportional but play different roles. See k·p perturbation theory for an example. Crystal momentum is like a wave envelope that describes how the wave varies from one unit cell to the next, but does not give any information about how the wave varies within each unit cell.

When k relates to crystal momentum instead of true momentum, the concept of k-space is still meaningful and extremely useful, but it differs in several ways from the non-crystal k-space discussed above. For example, in a crystal's k-space, there is an infinite set of points called the reciprocal lattice which are "equivalent" to k = 0 (this is analogous to aliasing). Likewise, the "first Brillouin zone" is a finite volume of k-space, such that every possible k is "equivalent" to exactly one point in this region.

Many-minds interpretation

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Many-minds_interpretation

The many-minds interpretation of quantum mechanics extends the many-worlds interpretation by proposing that the distinction between worlds should be made at the level of the mind of an individual observer. The concept was first introduced in 1970 by H. Dieter Zeh as a variant of the Hugh Everett interpretation in connection with quantum decoherence, and later (in 1981) explicitly called a many or multi-consciousness interpretation. The name many-minds interpretation was first used by David Albert and Barry Loewer in 1988.

History

Interpretations of quantum mechanics

Main article: Interpretations of quantum mechanics

The various interpretations of quantum mechanics typically involve explaining the mathematical formalism of quantum mechanics, or to create a physical picture of the theory. While the mathematical structure has a strong foundation, there is still much debate about the physical and philosophical interpretation of the theory. These interpretations aim to tackle various concepts such as:

1. Evolution of the state of a quantum system (given by the wavefunction), typically through the use of the Schrödinger equation. This concept is almost universally accepted, and is rarely put to debate.
2. The measurement problem, which relates to what is called wavefunction collapse – the collapse of a quantum state into a definite measurement (i.e. a specific eigenstate of the wavefunction). The debate on whether this collapse actually occurs is a central problem in interpreting quantum mechanics.

The standard solution to the measurement problem is the "Orthodox" or "Copenhagen" interpretation, which claims that the wave function collapses as the result of a measurement by an observer or apparatus external to the quantum system. An alternative interpretation, the Many-worlds Interpretation, was first described by Hugh Everett in 1957 (where it was called the relative state interpretation, the name Many-worlds was coined by Bryce Seligman DeWitt starting in the 1960s and finalized in the 1970s). His formalism of quantum mechanics denied that a measurement requires a wave collapse, instead suggesting that all that is truly necessary of a measurement is that a quantum connection is formed between the particle, the measuring device, and the observer.

The many-worlds interpretation

Main article: Many-worlds interpretation

In the original relative state formulation, Everett proposed that there is one universal wavefunction that describes the objective reality of the whole universe. He stated that when subsystems interact, the total system becomes a superposition of these subsystems. This includes observers and measurement systems, which become part of one universal state (the wavefunction) that is always described via the Schrödinger Equation (or its relativistic alternative). That is, the states of the subsystems that interacted become "entangled" in such a way that any definition of one must necessarily involve the other. Thus, each subsystem's state can only be described relative to each subsystem with which it interacts (hence the name relative state).

Everett suggested that the universe is actually indeterminate as a whole. For example, consider an observer measuring some particle that starts in an undetermined state, as both spin-up and spin-down, that is – a superposition of both possibilities. When an observer measures that particle's spin, however, it always registers as either up or down. The problem of how to understand this sudden shift from "both up and down" to "either up or down" is called the Measurement problem. According to the many-worlds interpretation, the act of measurement forced a “splitting” of the universe into two states, one spin-up and the other spin-down, and the two branches that extend from those two subsequently independent states. One branch measures up. The other measures down. Looking at the instrument informs the observer which branch he is on, but the system itself is indeterminate at this and, by logical extension, presumably any higher level.

The “worlds” in the many worlds theory is then just the complete measurement history up until and during the measurement in question, where splitting happens. These “worlds” each describe a different state of the universal wave function and cannot communicate. There is no collapse of the wavefunction into one state or another, but rather an observer finds itself in the world leading up to what measurement it has made and is unaware of the other possibilities that are equally real.

The many-minds interpretation

The many-minds interpretation of quantum theory is many-worlds with the distinction between worlds constructed at the level of the individual observer. Rather than the worlds that branch, it is the observer's mind that branches.

The problem with this interpretation is that it implies the observer must be in a superposition with herself, and that seems strange. In their 1988 paper, Albert and Loewer argued that the mind of an observer cannot be in an indefinite state because an observer must answer the question about which state of a system he has observed with complete certainty. If the observer's mind were in a superposition of states, then it could not attain such certainty. To overcome this contradiction, they suggest that a mind must always be in a definite state and only the “bodies” of the minds are in a superposition.

Accordingly, when an observer measures a quantum system and becomes entangled with it, the result is a larger quantum system. In regards to each possibility within this greater wave function, a mental state of the brain corresponds. Ultimately, only one of these mental states is experienced, leading the others to branch off and become inaccessible, albeit real. In this way, every sentient being possesses an infinity of minds, whose prevalence correspond to the amplitude of the wavefunction. As an observer checks a measurement, the probability of realizing a specific measurement directly correlates to the number of minds they have where they see that measurement. It is in this way that the probabilistic nature of quantum measurements are obtained by the Many-minds Interpretation.

Quantum non-locality in the many-minds interpretation

The body remains in an indeterminate state while the minds picks a stochastic result.

Consider an experiment that measures the polarization of two photons. When the photon is created, it has an indeterminate polarization. If a stream of these photons is passed through a polarization filter, 50% of the light is passed through. This corresponds to each photon having a 50% chance of aligning with the filter and thus passing, or being misaligned (by 90 degrees relative to the polarization filter) and being absorbed. Quantum mechanically, this means the photon is in a superposition of states where it is either passed or absorbed. Now, consider the inclusion of another photon and polarization detector. Now, the photons are created in such a way that they are entangled. That is, when one photon takes on a polarization state, the other photon will always behave as if it has the same polarization. For simplicity, take the second filter to either be perfectly aligned with the first, or to be perfectly misaligned (90 degree difference in angle, such that it is absorbed). If the detectors are aligned, both photons are passed (i.e. they are said to agree). If they are misaligned, only the first passes and the second is absorbed (now they disagree). Thus, the entanglement causes perfect correlations between the two measurements – regardless of separation distance, making the interaction non-local. This sort of experiment is further explained in Tim Maudlin's Quantum Non-Locality and Relativity, and can be related to Bell test experiments. Now, consider the analysis of this experiment from the many minds point of view:

No sentient observer

Consider the case where there is no sentient observer, i.e. no mind present to observe the experiment. In this case, the detector will be in an indefinite state. The photon is both passed and absorbed, and will remain in this state. The correlations are withheld in that none of the possible "minds", or wave function states, correspond to non correlated results.

One sentient observer

Now expand the situation to have one sentient being observing the device. Now, they too enter the indefinite state. Their eyes, body, and brain are seeing both spins at the same time. The mind however, stochastically chooses one of the directions, and that is what the mind sees. When this observer views the second detector, their body will see both results. Their mind will choose the result that agrees with the first detector, and the observer will see the expected results. However, the observer's mind seeing one result does not directly affect the distant state – there is just no wave function in which the expected correlations do not exist. The true correlation only happens when they actually view the second detector.

Two sentient observers

When two people look at two different detectors that scan entangled particles, both observers will enter an indefinite state, as with one observer. These results need not agree – the second observer's mind does not have to have results that correlate with the first's. When one observer tells the results to the second observer, their two minds cannot communicate and thus will only interact with the other's body, which is still indefinite. When the second observer responds, his body will respond with whatever result agrees with the first observer's mind. This means that both observer's minds will be in a state of the wavefunction that always get the expected results, but individually their results could be different.

Non-locality of the many-minds interpretation

As we have thus seen, any correlations seen in the wavefunction of each observer's minds are only concrete after interaction between the different polarizers. The correlations on the level of individual minds correspond to the appearance of quantum non-locality (or equivalently, violation of Bell's inequality). So the many world is non-local, or it cannot explain EPR-GHZ correlations.

Support

There is currently no empirical evidence for the many-minds interpretation. However, there are theories that do not discredit the many-minds interpretation. In light of Bell's analysis of the consequences of quantum non-locality, empirical evidence is needed to avoid inventing novel fundamental concepts (hidden variables). Two different solutions of the measurement problem then appear conceivable: consciousness causes collapse or Everett's relative state interpretation. In both cases a (suitably modified) psycho-physical parallelism can be re-established.

If neural processes can be described and analyzed then some experiments could potentially be created to test whether affecting neural processes can have an effect on a quantum system. Speculation about the details of this awareness-local physical system coupling on a purely theoretical basis could occur, however experimentally searching for them through neurological and psychological studies would be ideal.

Objections

Nothing within quantum theory itself requires each possibility within a wave function to complement a mental state. As all physical states (i.e. brain states) are quantum states, their associated mental states should be also. Nonetheless, it is not what one experiences within physical reality. Albert and Loewer argue that the mind must be intrinsically different than the physical reality as described by quantum theory. Thereby, they reject type-identity physicalism in favour of a non-reductive stance. However, Lockwood saves materialism through the notion of supervenience of the mental on the physical.

Nonetheless, the many-minds interpretation does not solve the mindless hulks problem as a problem of supervenience. Mental states do not supervene on brain states as a given brain state is compatible with different configurations of mental states.

Another serious objection is that workers in no collapse interpretations have produced no more than elementary models based on the definite existence of specific measuring devices. They have assumed, for example, that the Hilbert space of the universe splits naturally into a tensor product structure compatible with the measurement under consideration. They have also assumed, even when describing the behaviour of macroscopic objects, that it is appropriate to employ models in which only a few dimensions of Hilbert space are used to describe all the relevant behaviour.

Furthermore, as the many-minds interpretation is corroborated by our experience of physical reality, a notion of many unseen worlds and its compatibility with other physical theories (i.e. the principle of the conservation of mass) is difficult to reconcile. According to Schrödinger's equation, the mass-energy of the combined observed system and measurement apparatus is the same before and after. However, with every measurement process (i.e. splitting), the total mass-energy would seemingly increase.

Peter J. Lewis argues that the many-minds interpretation of quantum mechanics has absurd implications for agents facing life-or-death decisions.

In general, the many-minds theory holds that a conscious being who observes the outcome of a random zero-sum experiment will evolve into two successors in different observer states, each of whom observes one of the possible outcomes. Moreover, the theory advises one to favour choices in such situations in proportion to the probability that they will bring good results to one's various successors. But in a life-or-death case like an observer getting into the box with Schrödinger's cat, the observer will only have one successor, since one of the outcomes will ensure the observers death. So it seems that the many-minds interpretation advises one to get in the box with the cat, since it is certain that one's only successor will emerge unharmed. See also quantum suicide and immortality.

Finally, it supposes that there is some physical distinction between a conscious observer and a non-conscious measuring device, so it seems to require eliminating the strong Church–Turing hypothesis or postulating a physical model for consciousness.

Gravitational shielding

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Gravitational_shielding 

The term gravitational shielding refers to a hypothetical process of shielding an object from the influence of a gravitational field. Such processes, if they existed, would have the effect of reducing the weight of an object. The shape of the shielded region would be similar to a shadow from the gravitational shield. For example, the shape of the shielded region above a disk would be conical. The height of the cone's apex above the disk would vary directly with the height of the shielding disk above the Earth. Experimental evidence to date indicates that no such effect exists. Gravitational shielding is considered to be a violation of the equivalence principle and therefore inconsistent with both Newtonian theory and general relativity.

The concept of gravity shielding is a common concept in science fiction literature, especially for space travel. One of the first and best known examples is the fictional gravity shielding substance cavorite that appears in H. G. Wells' classic 1901 novel The First Men in the Moon. Wells was promptly criticized for using it by Jules Verne.

Tests of the equivalence principle

As of 2008, no experiment was successful in detecting positive shielding results. To quantify the amount of shielding, at the beginning of 20th century Quirino Majorana suggested an extinction coefficient h that modifies Newton's gravitational force law as follows:

${\displaystyle F=\frac{GMm}{r^2}{e}^{-h\int \rho (r)dr}}$

The best laboratory measurements have established an upper bound limit for shielding of 4.3×10⁻¹⁵ m²/kg. The best estimate based on the most accurate gravity anomaly data during the 1997 solar eclipse has provided a new constraint on the shielding parameter 6×10⁻¹⁹ m²/kg. However, astronomical observations impose much more stringent limits. Based on lunar observations available in 1908, Poincaré established that h can be no greater than 10⁻¹⁸ m²/kg. Subsequently, this bound has been greatly improved. Eckhardt showed that lunar ranging data implies an upper bound of 10⁻²² m²/kg, and Williams, et al., have improved this to h = (3 ± 5)×10⁻²² m²/kg. Note that the value is smaller than the uncertainty. The consequence of the negative results of those experiments (which are in good agreement with the predictions of general relativity) is, that every theory which contains shielding effects like Le Sage's theory of gravitation, must reduce those effects to an undetectable level. For a review of the current experimental limits on possible gravitational shielding, see the survey article by Bertolami, et al. Also, for a discussion of recent observations during solar eclipses, see the paper by Unnikrishnan et al.

Majorana's experiments and Russell's criticism

Some shielding experiments were conducted in the early 20th century by Quirino Majorana. Majorana claimed to have measured positive shielding effects. Henry Norris Russell's analysis of the tidal forces showed that Majorana's positive results had nothing to do with gravitational shielding. To bring Majorana's experiments following the equivalence principle of General Relativity he proposed a model, in which the mass of a body is diminished by the proximity of another body, but he denied any connection between gravitational shielding and his proposal of mass variation. For another explanation of Majorana's experiments, see Coïsson et al. But Majorana's results could not be confirmed up to this day (see the section above) and Russell's mass variation theory, although meant as a modification of general relativity, is inconsistent with standard physics as well.

Minority views

The consensus view of the scientific community is that gravitational shielding does not exist, but there have been occasional investigations into this topic, such as the 1999 NASA-funded paper that reported negative results. Eugene Podkletnov claimed in two papers, one of which he later withdrew, that objects held above a magnetically-levitated, superconducting, rotating disc underwent a reduction of between 0.5 and 2% in weight. Theoreticians have attempted to reconcile Podkletnov's claims with quantum gravity theory. In 2006, a research group funded by ESA claimed to have created a similar device that demonstrated positive results for the production of gravitomagnetism, although it produced only 0.0001 g.

Electrets

In his 1976 paper, Electromagnetism and Gravitation, physicist Edward Teller discussed experimentation with electrets, or materials with a permanent electric dipole moment, near its transition point to discover the transition between dipole states. On July 9, 1997, William Rhodes, an inventor, made a posting on Usenet concerning a discovery of an antigravity effect related to electrets. Also, Dr. Martin Tajmar, a physicist and professor for Space Systems at the Dresden University of Technology has written a paper on propellantless propulsion and makes numerous references to electrets. A patent for a gravitational attenuating material that utilizes an organic based material was made by inventor Ronald J. Kita.

Einstein–Cartan theory

Einstein–Cartan theory seems to allow gravitational shielding.