Quantum spacetime

From Wikipedia, the free encyclopedia

In mathematical physics, the concept of quantum spacetime is a generalization of the usual concept of spacetime in which some variables that ordinarily commute are assumed not to commute and form a different Lie algebra. The choice of that algebra still varies from theory to theory. As a result of this change some variables that are usually continuous may become discrete. Often only such discrete variables are called "quantized"; usage varies. The idea of quantum spacetime was proposed in the early days of quantum theory by Heisenberg and Ivanenko as a way to eliminate infinities from quantum field theory. The germ of the idea passed from Heisenberg to Rudolf Peierls, who noted that electrons in a magnetic field can be regarded as moving in a quantum space-time, and to Robert Oppenheimer, who carried it to Hartland Snyder, who published the first concrete example. Snyder's Lie algebra was made simple by C. N. Yang in the same year.

Physical reasons have been given to believe that physical spacetime is a quantum spacetime. In quantum mechanics position and momentum variables ${\displaystyle x,p}$ are already noncommutative, obey the Heisenberg uncertainty principle, and are continuous. Because of the Heisenberg uncertainty relations, greater energy is needed to probe smaller distances. Ultimately, according to gravity theory, the probing particles form black holes that destroy what was to be measured. The process cannot be repeated, so it cannot be counted as a measurement. This limited measurability led many to expect that our usual picture of continuous commutative spacetime breaks down at Planck scale distances, if not sooner.

Again, physical spacetime is expected to be quantum because physical coordinates are already slightly noncommutative. The astronomical coordinates of a star are modified by gravitational fields between us and the star, as in the deflection of light by the sun, one of the classic tests of general relativity. Therefore, the coordinates actually depend on gravitational field variables. According to quantum theories of gravity these field variables do not commute; therefore coordinates that depend on them likely do not commute.

Both arguments are based on pure gravity and quantum theory, and they limit the measurement of time by the only time constant in pure quantum gravity, the Planck time. Our instruments, however, are not purely gravitational but are made of particles. They may set a more severe, larger, limit than the Planck time.

Quantum spacetimes are often described mathematically using the noncommutative geometry of Connes, quantum geometry, or quantum groups.

Any noncommutative algebra with at least four generators could be interpreted as a quantum spacetime, but the following desiderata have been suggested:

• Local Lorentz group and Poincaré group symmetries should be retained, possibly in a generalised form. Their generalization often takes the form of a quantum group acting on the quantum spacetime algebra.
• The algebra might plausibly arise in an effective description of quantum gravity effects in some regime of that theory. For example, a physical parameter ${\displaystyle \lambda }$, perhaps the Planck length, might control the deviation from commutative classical spacetime, so that ordinary Lorentzian spacetime arises as ${\displaystyle \lambda \to 0}$.
• There might be a notion of quantum differential calculus on the quantum spacetime algebra, compatible with the (quantum) symmetry and preferably reducing to the usual differential calculus as ${\displaystyle \lambda \to 0}$.

This would permit wave equations for particles and fields and facilitate predictions for experimental deviations from classical spacetime physics that can then be tested experimentally.

• The Lie algebra should be semisimple. This makes it easier to formulate a finite theory.

Several models were found in the 1990s more or less meeting most of the above criteria.

Bicrossproduct model spacetime

The bicrossproduct model spacetime was introduced by Shahn Majid and Henri Ruegg and has Lie algebra relations

${\displaystyle [x_{i},x_{j}]=0,\quad [x_{i},t]=i\lambda x_{i}}$

for the spatial variables ${\displaystyle x_{i}}$ and the time variable ${\displaystyle t}$. Here ${\displaystyle \lambda }$ has dimensions of time and is therefore expected to be something like the Planck time. The Poincaré group here is correspondingly deformed, now to a certain bicrossproduct quantum group with the following characteristic features. 

Orbits for the action of the Lorentz group on momentum space in the construction of the bicrossproduct model in units of ${\displaystyle \lambda ^{-1}}$. Mass-shell hyperboloids are `squashed' into a cylinder.

 

The momentum generators ${\displaystyle p_{i}}$ commute among themselves but addition of momenta, reflected in the quantum group structure, is deformed (momentum space becomes a non-abelian group). Meanwhile, the Lorentz group generators enjoy their usual relations among themselves but act non-linearly on the momentum space. The orbits for this action are depicted in the figure as a cross-section of ${\displaystyle p_{0}}$ against one of the ${\displaystyle p_{i}}$. The on-shell region describing particles in the upper center of the image would normally be hyperboloids but these are now `squashed' into the cylinder

${\displaystyle {\sqrt {p_{1}^{2}+p_{2}^{2}+p_{3}^{2}}}<\lambda ^{-1}\,}$

in simplified units. The upshot is that Lorentz-boosting a momentum will never increase it above the Planck momentum. The existence of a highest momentum scale or lowest distance scale fits the physical picture. This squashing comes from the non-linearity of the Lorentz boost and is an endemic feature of bicrossproduct quantum groups known since their introduction in 1988. Some physicists dub the bicrossproduct model doubly special relativity, since it sets an upper limit to both speed and momentum. 

Another consequence of the squashing is that the propagation of particles is deformed, even of light, leading to a variable speed of light. This prediction requires the particular ${\displaystyle p_{0},p_{i}}$ to be the physical energy and spatial momentum (as opposed to some other function of them). Arguments for this identification were provided in 1999 by Giovanni Amelino-Camelia and Majid through a study of plane waves for a quantum differential calculus in the model. They take the form

${\displaystyle e^{i\sum _{i}p_{i}x_{i}}e^{ip_{0}t}\,}$

in other words a form which is sufficiently close to classical that one might plausibly believe the interpretation. At the moment such wave analysis represents the best hope to obtain physically testable predictions from the model. 

Prior to this work there were a number of unsupported claims to make predictions from the model based solely on the form of the Poincaré quantum group. There were also claims based on an earlier ${\displaystyle \kappa }$-Poincaré quantum group introduced by Jurek Lukierski and co-workers which should be viewed as an important precursor to the bicrossproduct one, albeit without the actual quantum spacetime and with different proposed generators for which the above picture does not apply. The bicrossproduct model spacetime has also been called ${\displaystyle \kappa }$-deformed spacetime with ${\displaystyle \kappa =\lambda ^{-1}}$.

q-Deformed spacetime

This model was introduced independently by a team working under Julius Wess in 1990 and by Majid and coworkers in a series of papers on braided matrices starting a year later. The point of view in the second approach is that usual Minkowski spacetime has a nice description via Pauli matrices as the space of 2 x 2 hermitian matrices. In quantum group theory and using braided monoidal category methods one has a natural q-version of this defined here for real values of ${\displaystyle q}$ as a `braided hermitian matrix' of generators and relations

${\displaystyle {\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}={\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}^{\dagger },\quad \beta \alpha =q^{2}\alpha \beta ,\ [\alpha ,\delta ]=0,\ [\beta ,\gamma ]=(1-q^{-2})\alpha (\delta -\alpha ),\ [\delta ,\beta ]=(1-q^{-2})\alpha \beta }$

These relations say that the generators commute as ${\displaystyle q\to 1}$ thereby recovering usual Minkowski space. One can work with more familiar variables ${\displaystyle x,y,z,t}$ as linear combinations of these. In particular, time

${\displaystyle t={\text{Trace}}_{q}{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}=q\delta +q^{-1}\alpha }$

is given by a natural braided trace of the matrix and commutes with the other generators (so this model has a very different flavour from the bicrossproduct one). The braided-matrix picture also leads naturally to a quantity

${\displaystyle {\det }_{q}{\begin{pmatrix}\alpha &\beta \\\gamma &\delta \end{pmatrix}}=\alpha \delta -q^{2}\gamma \beta }$

which as ${\displaystyle q\to 1}$ returns us the usual Minkowski distance (this translates to a metric in the quantum differential geometry). The parameter ${\displaystyle q=e^{\lambda }}$ or ${\displaystyle q=e^{i\lambda }}$ is dimensionless and ${\displaystyle \lambda }$ is thought to be a ratio of the Planck scale and the cosmological length. That is, there are indications that this model relates to quantum gravity with non-zero cosmological constant, the choice of ${\displaystyle q}$ depending on whether this is positive or negative. We have described the mathematically better understood but perhaps less physically justified positive case here. 

A full understanding of this model requires (and was concurrent with the development of) a full theory of `braided linear algebra' for such spaces. The momentum space for the theory is another copy of the same algebra and there is a certain `braided addition' of momentum on it expressed as the structure of a braided Hopf algebra or quantum group in a certain braided monoidal category). This theory by 1993 had provided the corresponding ${\displaystyle q}$-deformed Poincaré group as generated by such translations and ${\displaystyle q}$-Lorentz transformations, completing the interpretation as a quantum spacetime.

In the process it was discovered that the Poincaré group not only had to be deformed but had to be extended to include dilations of the quantum spacetime. For such a theory to be exact we would need all particles in the theory to be massless, which is consistent with experiment as masses of elementary particles are indeed vanishingly small compared to the Planck mass. If current thinking in cosmology is correct then this model is more appropriate, but it is significantly more complicated and for this reason its physical predictions have yet to be worked out.

Fuzzy or spin model spacetime

This refers in modern usage to the angular momentum algebra

${\displaystyle [x_{1},x_{2}]=2i\lambda x_{3},\ [x_{2},x_{3}]=2i\lambda x_{1},\ [x_{3},x_{1}]=2i\lambda x_{2}}$

familiar from quantum mechanics but interpreted in this context as coordinates of a quantum space or spacetime. These relations were proposed by Roger Penrose in his earliest spin network theory of space. It is a toy model of quantum gravity in 3 spacetime dimensions (not the physical 4) with a Euclidean (not the physical Minkowskian) signature. It was again proposed in this context by Gerardus 't Hooft. A further development including a quantum differential calculus and an action of a certain `quantum double' quantum group as deformed Euclidean group of motions was given by Majid and E. Batista.

A striking feature of the noncommutative geometry here is that the smallest covariant quantum differential calculus has one dimension higher than expected, namely 4, suggesting that the above can also be viewed as the spatial part of a 4-dimensional quantum spacetime. The model should not be confused with fuzzy spheres which are finite-dimensional matrix algebras which one can think of as spheres in the spin model spacetime of fixed radius.

Heisenberg model spacetimes

The quantum spacetime of Hartland Snyder proposes that

${\displaystyle [x_{\mu },x_{\nu }]=iM_{\mu \nu }}$

where the ${\displaystyle M_{\mu \nu }}$ generate the Lorentz group. This quantum spacetime and that of C. N. Yang entail a radical unification of spacetime, energy-momentum, and angular momentum. 

The idea was revived in a modern context by Sergio Doplicher, Klaus Fredenhagen and John Roberts in 1995 by letting ${\displaystyle M_{\mu \nu }}$ simply be viewed as some function of ${\displaystyle x_{\mu }}$ as defined by the above relation, and any relations involving it viewed as higher order relations among the ${\displaystyle x_{\mu }}$. The Lorentz symmetry is arranged so as to transform the indices as usual and without being deformed.

An even simpler variant of this model is to let ${\displaystyle M}$ here be a numerical antisymmetric tensor, in which context it is usually denoted ${\displaystyle \theta }$, so the relations are ${\displaystyle [x_{\mu },x_{\nu }]=i\theta _{\mu \nu }}$. In even dimensions ${\displaystyle D}$, any nondegenerate such theta can be transformed to a normal form in which this really is just the Heisenberg algebra but the difference that the variables are being proposed as those of spacetime. This proposal was for a time quite popular because of its familiar form of relations and because it has been argued that it emerges from the theory of open strings landing on D-branes, see noncommutative quantum field theory and Moyal plane. However, it should be realised that this D-brane lives in some of the higher spacetime dimensions in the theory and hence it is not our physical spacetime that string theory suggests to be effectively quantum in this way. You also have to subscribe to D-branes as an approach to quantum gravity in the first place. Even when posited as quantum spacetime it is hard to obtain physical predictions and one reason for this is that if ${\displaystyle \theta }$ is a tensor then by dimensional analysis it should have dimensions of length${\displaystyle {}^{2}}$, and if this length is speculated to be the Planck length then the effects would be even harder to ever detect than for other models.

Noncommutative extensions to spacetime

Although not quantum spacetime in the sense above, another use of noncommutative geometry is to tack on `noncommutative extra dimensions' at each point of ordinary spacetime. Instead of invisible curled up extra dimensions as in string theory, Alain Connes and coworkers have argued that the coordinate algebra of this extra part should be replaced by a finite-dimensional noncommutative algebra. For a certain reasonable choice of this algebra, its representation and extended Dirac operator, one is able to recover the Standard Model of elementary particles. In this point of view the different kinds of matter particles are manifestations of geometry in these extra noncommutative directions. Connes's first works here date from 1989 but has been developed considerably since then. Such an approach can theoretically be combined with quantum spacetime as above.

Quantum superposition

From Wikipedia, the free encyclopedia

Quantum superposition of states and decoherence

 

Quantum superposition is a fundamental principle of quantum mechanics. It states that, much like waves in classical physics, any two (or more) quantum states can be added together ("superposed") and the result will be another valid quantum state; and conversely, that every quantum state can be represented as a sum of two or more other distinct states. Mathematically, it refers to a property of solutions to the Schrödinger equation; since the Schrödinger equation is linear, any linear combination of solutions will also be a solution.

An example of a physically observable manifestation of the wave nature of quantum systems is the interference peaks from an electron beam in a double-slit experiment. The pattern is very similar to the one obtained by diffraction of classical waves.

Another example is a quantum logical qubit state, as used in quantum information processing, which is a quantum superposition of the "basis states" ${\displaystyle |0\rangle }$ and ${\displaystyle |1\rangle }$. Here ${\displaystyle |0\rangle }$ is the Dirac notation for the quantum state that will always give the result 0 when converted to classical logic by a measurement. Likewise ${\displaystyle |1\rangle }$ is the state that will always convert to 1. Contrary to a classical bit that can only be in the state corresponding to 0 or the state corresponding to 1, a qubit may be in a superposition of both states. This means that the probabilities of measuring 0 or 1 for a qubit are in general neither 0.0 nor 1.0, and multiple measurements made on qubits in identical states will not always give the same result.

Concept

The principle of quantum superposition states that if a physical system may be in one of many configurations—arrangements of particles or fields—then the most general state is a combination of all of these possibilities, where the amount in each configuration is specified by a complex number. 

For example, if there are two configurations labelled by 0 and 1, the most general state would be

${\displaystyle c_{0}\mid 0\rangle +c_{1}\mid 1\rangle }$

where the coefficients are complex numbers describing how much goes into each configuration.

The principle was described by Paul Dirac as follows:

The general principle of superposition of quantum mechanics applies to the states [that are theoretically possible without mutual interference or contradiction] ... of any one dynamical system. It requires us to assume that between these states there exist peculiar relationships such that whenever the system is definitely in one state we can consider it as being partly in each of two or more other states. The original state must be regarded as the result of a kind of superposition of the two or more new states, in a way that cannot be conceived on classical ideas. Any state may be considered as the result of a superposition of two or more other states, and indeed in an infinite number of ways. Conversely, any two or more states may be superposed to give a new state...

The non-classical nature of the superposition process is brought out clearly if we consider the superposition of two states, A and B, such that there exists an observation which, when made on the system in state A, is certain to lead to one particular result, a say, and when made on the system in state B is certain to lead to some different result, b say. What will be the result of the observation when made on the system in the superposed state? The answer is that the result will be sometimes a and sometimes b, according to a probability law depending on the relative weights of A and B in the superposition process. It will never be different from both a and b [i.e, either a or b]. The intermediate character of the state formed by superposition thus expresses itself through the probability of a particular result for an observation being intermediate between the corresponding probabilities for the original states, not through the result itself being intermediate between the corresponding results for the original states.

Anton Zeilinger, referring to the prototypical example of the double-slit experiment, has elaborated regarding the creation and destruction of quantum superposition:

[T]he superposition of amplitudes ... is only valid if there is no way to know, even in principle, which path the particle took. It is important to realize that this does not imply that an observer actually takes note of what happens. It is sufficient to destroy the interference pattern, if the path information is accessible in principle from the experiment or even if it is dispersed in the environment and beyond any technical possibility to be recovered, but in principle still ‘‘out there.’’ The absence of any such information is the essential criterion for quantum interference to appear.

Theory

Examples

For an equation describing a physical phenomenon, the superposition principle states that a combination of solutions to a linear equation is also a solution of it. When this is true the equation is said to obey the superposition principle. Thus, if state vectors f₁, f₂ and f₃ each solve the linear equation on ψ, then ψ = c₁ f₁ + c₂ f₂ + c₃ f₃ would also be a solution, in which each c is a coefficient. The Schrödinger equation is linear, so quantum mechanics follows this. 

For example, consider an electron with two possible configurations, up and down. This describes the physical system of a qubit.

${\displaystyle c_{1}\mid \uparrow \rangle +c_{2}\mid \downarrow \rangle }$

is the most general state. But these coefficients dictate probabilities for the system to be in either configuration. The probability for a specified configuration is given by the square of the absolute value of the coefficient. So the probabilities should add up to 1. The electron is in one of those two states for sure.

${\displaystyle p_{\text{up}}=\mid c_{1}\mid ^{2}}$

${\displaystyle p_{\text{down}}=\mid c_{2}\mid ^{2}}$

${\displaystyle p_{\text{up or down}}=p_{\text{up}}+p_{\text{down}}=1}$

Continuing with this example: If a particle can be in state  up and  down, it can also be in a state where it is an amount 3i/5 in up and an amount 4/5 in down.

${\displaystyle |\psi \rangle ={3 \over 5}i|\uparrow \rangle +{4 \over 5}|\downarrow \rangle .}$

In this, the probability for up is ${\displaystyle \left|\;{\frac {3i}{5}}\;\right|^{2}={\frac {9}{25}}}$. The probability for down is ${\displaystyle \left|\;{\frac {4}{5}}\;\right|^{2}={\frac {16}{25}}}$. Note that ${\displaystyle {\frac {9}{25}}+{\frac {16}{25}}=1}$. 

In the description, only the relative size of the different components matter, and their angle to each other on the complex plane. This is usually stated by declaring that two states which are a multiple of one another are the same as far as the description of the situation is concerned. Either of these describe the same state for any nonzero ${\displaystyle \alpha }$

${\displaystyle |\psi \rangle \approx \alpha |\psi \rangle }$

The fundamental law of quantum mechanics is that the evolution is linear, meaning that if state A turns into A′ and B turns into B′ after 10 seconds, then after 10 seconds the superposition ${\displaystyle \psi }$ turns into a mixture of A′ and B′ with the same coefficients as A and B. 

For example, if we have the following

${\displaystyle \mid \uparrow \rangle \to \mid \downarrow \rangle }$

${\displaystyle \mid \downarrow \rangle \to {\frac {3i}{5}}\mid \uparrow \rangle +{\frac {4}{5}}\mid \downarrow \rangle }$

Then after those 10 seconds our state will change to

${\displaystyle c_{1}\mid \uparrow \rangle +c_{2}\mid \downarrow \rangle \to c_{1}\left(\mid \downarrow \rangle \right)+c_{2}\left({\frac {3i}{5}}\mid \uparrow \rangle +{\frac {4}{5}}\mid \downarrow \rangle \right)}$

So far there have just been 2 configurations, but there can be infinitely many. 

In illustration, a particle can have any position, so that there are different configurations which have any value of the position x. These are written:

${\displaystyle |x\rangle }$

The principle of superposition guarantees that there are states which are arbitrary superpositions of all the positions with complex coefficients:

${\displaystyle \sum _{x}\psi (x)|x\rangle }$

This sum is defined only if the index x is discrete. If the index is over ${\displaystyle \mathbb {R} }$, then the sum is replaced by an integral. The quantity ${\displaystyle \psi (x)}$ is called the wave function of the particle. 

If we consider a qubit with both position and spin, the state is a superposition of all possibilities for both:

${\displaystyle \sum _{x}\psi _{+}(x)|x,\uparrow \rangle +\psi _{-}(x)|x,\downarrow \rangle \,}$

The configuration space of a quantum mechanical system cannot be worked out without some physical knowledge. The input is usually the allowed different classical configurations, but without the duplication of including both position and momentum.

A pair of particles can be in any combination of pairs of positions. A state where one particle is at position x and the other is at position y is written ${\displaystyle |x,y\rangle }$. The most general state is a superposition of the possibilities:

${\displaystyle \sum _{xy}A(x,y)|x,y\rangle \,}$

The description of the two particles is much larger than the description of one particle—it is a function in twice the number of dimensions. This is also true in probability, when the statistics of two random variables are correlated. If two particles are uncorrelated, the probability distribution for their joint position P(x, y) is a product of the probability of finding one at one position and the other at the other position:

${\displaystyle P(x,y)=P_{x}(x)P_{y}(y)\,}$

In quantum mechanics, two particles can be in special states where the amplitudes of their position are uncorrelated. For quantum amplitudes, the word entanglement replaces the word correlation, but the analogy is exact. A disentangled wave function has the form:

${\displaystyle A(x,y)=\psi _{x}(x)\psi _{y}(y)\,}$

while an entangled wavefunction does not have this form.

Analogy with probability

In probability theory there is a similar principle. If a system has a probabilistic description, this description gives the probability of any configuration, and given any two different configurations, there is a state which is partly this and partly that, with positive real number coefficients, the probabilities, which say how much of each there is. 

For example, if we have a probability distribution for where a particle is, it is described by the "state"

${\displaystyle \sum _{x}\rho (x)|x\rangle }$

Where ${\displaystyle \rho }$ is the probability density function, a positive number that measures the probability that the particle will be found at a certain location.

The evolution equation is also linear in probability, for fundamental reasons. If the particle has some probability for going from position x to y, and from z to y, the probability of going to y starting from a state which is half-x and half-z is a half-and-half mixture of the probability of going to y from each of the options. This is the principle of linear superposition in probability.

Quantum mechanics is different, because the numbers can be positive or negative. While the complex nature of the numbers is just a doubling, if you consider the real and imaginary parts separately, the sign of the coefficients is important. In probability, two different possible outcomes always add together, so that if there are more options to get to a point z, the probability always goes up. In quantum mechanics, different possibilities can cancel.

In probability theory with a finite number of states, the probabilities can always be multiplied by a positive number to make their sum equal to one. For example, if there is a three state probability system:

${\displaystyle x|1\rangle +y|2\rangle +z|3\rangle \,}$

where the probabilities ${\displaystyle x,y,z}$ are positive numbers. Rescaling x,y,z so that

${\displaystyle x+y+z=1\,}$

The geometry of the state space is a revealed to be a triangle. In general it is a simplex. There are special points in a triangle or simplex corresponding to the corners, and these points are those where one of the probabilities is equal to 1 and the others are zero. These are the unique locations where the position is known with certainty.

In a quantum mechanical system with three states, the quantum mechanical wavefunction is a superposition of states again, but this time twice as many quantities with no restriction on the sign:

${\displaystyle A|1\rangle +B|2\rangle +C|3\rangle =(A_{r}+iA_{i})|1\rangle +(B_{r}+iB_{i})|2\rangle +(C_{r}+iC_{i})|3\rangle \,}$

rescaling the variables so that the sum of the squares is 1, the geometry of the space is revealed to be a high-dimensional sphere

${\displaystyle A_{r}^{2}+A_{i}^{2}+B_{r}^{2}+B_{i}^{2}+C_{r}^{2}+C_{i}^{2}=1\,}$.

A sphere has a large amount of symmetry, it can be viewed in different coordinate systems or bases. So unlike a probability theory, a quantum theory has a large number of different bases in which it can be equally well described. The geometry of the phase space can be viewed as a hint that the quantity in quantum mechanics which corresponds to the probability is the absolute square of the coefficient of the superposition.

Hamiltonian evolution

The numbers that describe the amplitudes for different possibilities define the kinematics, the space of different states. The dynamics describes how these numbers change with time. For a particle that can be in any one of infinitely many discrete positions, a particle on a lattice, the superposition principle tells you how to make a state:

${\displaystyle \sum _{n}\psi _{n}|n\rangle \,}$

So that the infinite list of amplitudes ${\textstyle (\ldots ,\psi _{-2},\psi _{-1},\psi _{0},\psi _{1},\psi _{2},\ldots )}$ completely describes the quantum state of the particle. This list is called the state vector, and formally it is an element of a Hilbert space, an infinite-dimensional complex vector space. It is usual to represent the state so that the sum of the absolute squares of the amplitudes is one:

${\displaystyle \sum \psi _{n}^{*}\psi _{n}=1}$

For a particle described by probability theory random walking on a line, the analogous thing is the list of probabilities ${\textstyle (\ldots ,P_{-2},P_{-1},P_{0},P_{1},P_{2},\ldots )}$, which give the probability of any position. The quantities that describe how they change in time are the transition probabilities ${\displaystyle \scriptstyle K_{x\rightarrow y}(t)}$, which gives the probability that, starting at x, the particle ends up at y time t later. The total probability of ending up at y is given by the sum over all the possibilities

${\displaystyle P_{y}(t_{0}+t)=\sum _{x}P_{x}(t_{0})K_{x\rightarrow y}(t)\,}$

The condition of conservation of probability states that starting at any x, the total probability to end up somewhere must add up to 1:

${\displaystyle \sum _{y}K_{x\rightarrow y}=1\,}$

So that the total probability will be preserved, K is what is called a stochastic matrix. 

When no time passes, nothing changes: for 0 elapsed time ${\displaystyle \scriptstyle K{x\rightarrow y}(0)=\delta _{xy}}$, the K matrix is zero except from a state to itself. So in the case that the time is short, it is better to talk about the rate of change of the probability instead of the absolute change in the probability.

${\displaystyle P_{y}(t+dt)=P_{y}(t)+dt\,\sum _{x}P_{x}R_{x\rightarrow y}\,}$

where ${\displaystyle \scriptstyle R_{x\rightarrow y}}$ is the time derivative of the K matrix:

${\displaystyle R_{x\rightarrow y}={K_{x\rightarrow y}\,dt-\delta _{xy} \over dt}.\,}$

The equation for the probabilities is a differential equation that is sometimes called the master equation:

${\displaystyle {dP_{y} \over dt}=\sum _{x}P_{x}R_{x\rightarrow y}\,}$

The R matrix is the probability per unit time for the particle to make a transition from x to y. The condition that the K matrix elements add up to one becomes the condition that the R matrix elements add up to zero:

${\displaystyle \sum _{y}R_{x\rightarrow y}=0\,}$

One simple case to study is when the R matrix has an equal probability to go one unit to the left or to the right, describing a particle that has a constant rate of random walking. In this case ${\displaystyle \scriptstyle R_{x\rightarrow y}}$ is zero unless y is either x + 1, x, or x − 1, when y is x + 1 or x − 1, the R matrix has value c, and in order for the sum of the R matrix coefficients to equal zero, the value of ${\displaystyle R_{x\rightarrow x}}$ must be −2c. So the probabilities obey the discretized diffusion equation:

${\displaystyle {dP_{x} \over dt}=c(P_{x+1}-2P_{x}+P_{x-1})\,}$

which, when c is scaled appropriately and the P distribution is smooth enough to think of the system in a continuum limit becomes:

${\displaystyle {\partial P(x,t) \over \partial t}=c{\partial ^{2}P \over \partial x^{2}}\,}$

Which is the diffusion equation. 

Quantum amplitudes give the rate at which amplitudes change in time, and they are mathematically exactly the same except that they are complex numbers. The analog of the finite time K matrix is called the U matrix:

${\displaystyle \psi _{n}(t)=\sum _{m}U_{nm}(t)\psi _{m}\,}$

Since the sum of the absolute squares of the amplitudes must be constant, ${\displaystyle U}$ must be unitary:

${\displaystyle \sum _{n}U_{nm}^{*}U_{np}=\delta _{mp}\,}$

or, in matrix notation,

${\displaystyle U^{\dagger }U=I\,}$

The rate of change of U is called the Hamiltonian H, up to a traditional factor of i:

${\displaystyle H_{mn}=i{d \over dt}U_{mn}}$

The Hamiltonian gives the rate at which the particle has an amplitude to go from m to n. The reason it is multiplied by i is that the condition that U is unitary translates to the condition:

${\displaystyle (I+iH^{\dagger }\,dt)(I-iH\,dt)=I}$

${\displaystyle H^{\dagger }-H=0\,}$

which says that H is Hermitian. The eigenvalues of the Hermitian matrix H are real quantities, which have a physical interpretation as energy levels. If the factor i were absent, the H matrix would be antihermitian and would have purely imaginary eigenvalues, which is not the traditional way quantum mechanics represents observable quantities like the energy. 

For a particle that has equal amplitude to move left and right, the Hermitian matrix H is zero except for nearest neighbors, where it has the value c. If the coefficient is everywhere constant, the condition that H is Hermitian demands that the amplitude to move to the left is the complex conjugate of the amplitude to move to the right. The equation of motion for ${\displaystyle \psi }$ is the time differential equation:

${\displaystyle i{d\psi _{n} \over dt}=c^{*}\psi _{n+1}+c\psi _{n-1}}$

In the case in which left and right are symmetric, c is real. By redefining the phase of the wavefunction in time, ${\displaystyle \psi \rightarrow \psi e^{i2ct}}$, the amplitudes for being at different locations are only rescaled, so that the physical situation is unchanged. But this phase rotation introduces a linear term.

${\displaystyle i{d\psi _{n} \over dt}=c\psi _{n+1}-2c\psi _{n}+c\psi _{n-1},}$

which is the right choice of phase to take the continuum limit. When ${\displaystyle c}$ is very large and ${\displaystyle \psi }$ is slowly varying so that the lattice can be thought of as a line, this becomes the free Schrödinger equation:

${\displaystyle i{\partial \psi \over \partial t}=-{\partial ^{2}\psi \over \partial x^{2}}}$

If there is an additional term in the H matrix that is an extra phase rotation that varies from point to point, the continuum limit is the Schrödinger equation with a potential energy:

${\displaystyle i{\partial \psi \over \partial t}=-{\partial ^{2}\psi \over \partial x^{2}}+V(x)\psi }$

These equations describe the motion of a single particle in non-relativistic quantum mechanics.

Quantum mechanics in imaginary time

The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step ${\displaystyle \scriptstyle K_{m\rightarrow n}}$, the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:

${\displaystyle K_{x\rightarrow y}(T)=\sum _{x(t)}\prod _{t}K_{x(t)x(t+1)}\,}$

where the sum extends over all paths ${\displaystyle x(t)}$ with the property that ${\displaystyle x(0)=0}$ and ${\displaystyle x(T)=y}$. The analogous expression in quantum mechanics is the path integral. 

A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys detailed balance when the stationary distribution ${\displaystyle \rho _{n}}$ has the property:

${\displaystyle \rho _{n}K_{n\rightarrow m}=\rho _{m}K_{m\rightarrow n}\,}$

Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m ${\displaystyle \rho _{m}}$ times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.

${\displaystyle \rho _{n}R_{n\rightarrow m}=\rho _{m}R_{m\rightarrow n}\,}$

When the R matrix obeys detailed balance, the scale of the probabilities can be redefined using the stationary distribution so that they no longer sum to 1:

${\displaystyle p'_{n}={\sqrt {\rho _{n}}}\;p_{n}\,}$

In the new coordinates, the R matrix is rescaled as follows:

${\displaystyle {\sqrt {\rho _{n}}}R_{n\rightarrow m}{1 \over {\sqrt {\rho _{m}}}}=H_{nm}\,}$

and H is symmetric

${\displaystyle H_{nm}=H_{mn}\,}$

This matrix H defines a quantum mechanical system:

${\displaystyle i{d \over dt}\psi _{n}=\sum H_{nm}\psi _{m}\,}$

whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The eigenvectors are the same too, except expressed in the rescaled basis. The stationary distribution of the statistical system is the ground state of the Hamiltonian and it has energy exactly zero, while all the other energies are positive. If H is exponentiated to find the U matrix:

${\displaystyle U(t)=e^{-iHt}\,}$

and t is allowed to take on complex values, the K' matrix is found by taking time imaginary.

${\displaystyle K'(t)=e^{-Ht}\,}$

For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on supersymmetry.

Experiments and applications

Successful experiments involving superpositions of relatively large (by the standards of quantum physics) objects have been performed.

• A "cat state" has been achieved with photons.
• A beryllium ion has been trapped in a superposed state.
• A double slit experiment has been performed with molecules as large as buckyballs.
• A 2013 experiment superposed molecules containing 15,000 each of protons, neutrons and electrons. The molecules were of compounds selected for their good thermal stability, and were evaporated into a beam at a temperature of 600 K. The beam was prepared from highly purified chemical substances, but still contained a mixture of different molecular species. Each species of molecule interfered only with itself, as verified by mass spectrometry.
• An experiment involving a superconducting quantum interference device ("SQUID") has been linked to the theme of the "cat state" thought experiment.

By use of very low temperatures, very fine experimental arrangements were made to protect in near isolation and preserve the coherence of intermediate states, for a duration of time, between preparation and detection, of SQUID currents. Such a SQUID current is a coherent physical assembly of perhaps billions of electrons. Because of its coherence, such an assembly may be regarded as exhibiting "collective states" of a macroscopic quantal entity. For the principle of superposition, after it is prepared but before it is detected, it may be regarded as exhibiting an intermediate state. It is not a single-particle state such as is often considered in discussions of interference, for example by Dirac in his famous dictum stated above. Moreover, though the 'intermediate' state may be loosely regarded as such, it has not been produced as an output of a secondary quantum analyser that was fed a pure state from a primary analyzer, and so this is not an example of superposition as strictly and narrowly defined.

Nevertheless, after preparation, but before measurement, such a SQUID state may be regarded in a manner of speaking as a "pure" state that is a superposition of a clockwise and an anti-clockwise current state. In a SQUID, collective electron states can be physically prepared in near isolation, at very low temperatures, so as to result in protected coherent intermediate states. Remarkable here is that there are found two well-separated respectively self-coherent collective states that exhibit such metastability. The crowd of electrons tunnels back and forth between the clockwise and the anti-clockwise states, as opposed to forming a single intermediate state in which there is no definite collective sense of current flow.

• An experiment involving a flu virus has been proposed.
• A piezoelectric "tuning fork" has been constructed, which can be placed into a superposition of vibrating and non-vibrating states. The resonator comprises about 10 trillion atoms.
• Recent research indicates that chlorophyll within plants appears to exploit the feature of quantum superposition to achieve greater efficiency in transporting energy, allowing pigment proteins to be spaced further apart than would otherwise be possible.
• An experiment has been proposed, with a bacterial cell cooled to 10 mK, using an electromechanical oscillator. At that temperature, all metabolism would be stopped, and the cell might behave virtually as a definite chemical species. For detection of interference, it would be necessary that the cells be supplied in large numbers as pure samples of identical and detectably recognizable virtual chemical species. It is not known whether this requirement can be met by bacterial cells. They would be in a state of suspended animation during the experiment.

In quantum computing the phrase "cat state" often refers to the GHZ state, the special entanglement of qubits wherein the qubits are in an equal superposition of all being 0 and all being 1; i.e.,

${\displaystyle |\psi \rangle ={\frac {1}{\sqrt {2}}}{\bigg (}|00\ldots 0\rangle +|11\ldots 1\rangle {\bigg )}.}$

Formal interpretation

Applying the superposition principle to a quantum mechanical particle, the configurations of the particle are all positions, so the superpositions make a complex wave in space. The coefficients of the linear superposition are a wave which describes the particle as best as is possible, and whose amplitude interferes according to the Huygens principle. 

For any physical property in quantum mechanics, there is a list of all the states where that property has some value. These states are necessarily perpendicular to each other using the Euclidean notion of perpendicularity which comes from sums-of-squares length, except that they also must not be i multiples of each other. This list of perpendicular states has an associated value which is the value of the physical property. The superposition principle guarantees that any state can be written as a combination of states of this form with complex coefficients.

Write each state with the value q of the physical quantity as a vector in some basis ${\displaystyle \psi _{n}^{q}}$, a list of numbers at each value of n for the vector which has value q for the physical quantity. Now form the outer product of the vectors by multiplying all the vector components and add them with coefficients to make the matrix

${\displaystyle A_{nm}=\sum _{q}q\psi _{n}^{*q}\psi _{m}^{q}}$

where the sum extends over all possible values of q. This matrix is necessarily symmetric because it is formed from the orthogonal states, and has eigenvalues q. The matrix A is called the observable associated to the physical quantity. It has the property that the eigenvalues and eigenvectors determine the physical quantity and the states which have definite values for this quantity.

Every physical quantity has a Hermitian linear operator associated to it, and the states where the value of this physical quantity is definite are the eigenstates of this linear operator. The linear combination of two or more eigenstates results in quantum superposition of two or more values of the quantity. If the quantity is measured, the value of the physical quantity will be random, with a probability equal to the square of the coefficient of the superposition in the linear combination. Immediately after the measurement, the state will be given by the eigenvector corresponding to the measured eigenvalue.

Physical interpretation

It is natural to ask why ordinary everyday objects and events do not seem to display quantum mechanical features such as superposition. Indeed, this is sometimes regarded as "mysterious", for instance by Richard Feynman. In 1935, Erwin Schrödinger devised a well-known thought experiment, now known as Schrödinger's cat, which highlighted this dissonance between quantum mechanics and classical physics. One modern view is that this mystery is explained by quantum decoherence. A macroscopic system (such as a cat) may evolve over time into a superposition of classically distinct quantum states (such as "alive" and "dead"). However, the state of the cat is entangled with the state of its environment (for instance, the molecules in the atmosphere surrounding it). If one averages over the quantum states of the environment—a physically reasonable procedure unless the quantum state of all the particles making up the environment can be controlled or measured precisely—the resulting mixed quantum state for the cat is very close to a classical probabilistic state where the cat has some definite probability to be dead or alive, just as a classical observer would expect in this situation.

The Heisenberg uncertainty principle declares that for any given instant of time, the position and velocity of an electron or other subatomic particle cannot both be exactly determined, and that a state where one of them has a definite value corresponds to a superposition of many states for the other.