Imaginary time

Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics. It finds uses in connecting quantum mechanics with statistical mechanics and in certain cosmological theories.

Mathematically, imaginary time is real time which has undergone a Wick rotation so that its coordinates are multiplied by the imaginary root i. Imaginary time is not imaginary in the sense that it is unreal or made-up (any more than, say, irrational numbers defy logic), it is simply expressed in terms of what mathematicians call imaginary numbers.

Origins

Mathematically, imaginary time may be obtained from real time via a Wick rotation by in the complex plane: , where Is defined to be , and is known as the imaginary unit.

Stephen Hawking popularized the concept of imaginary time in his book The Universe in a Nutshell.

One might think this means that imaginary numbers are just a mathematical game having nothing to do with the real world. From the viewpoint of positivist philosophy, however, one cannot determine what is real. All one can do is find which mathematical models describe the universe we live in. It turns out that a mathematical model involving imaginary time predicts not only effects we have already observed but also effects we have not been able to measure yet nevertheless believe in for other reasons. So what is real and what is imaginary? Is the distinction just in our minds?

In fact, the names "real" and "imaginary" for numbers are just a historical accident, much like the names "rational" and "irrational":

...the words real and imaginary are picturesque relics of an age when the nature of complex numbers was not properly understood.

In cosmology

In the Minkowski spacetime model adopted by the theory of relativity, spacetime is represented as a four-dimensional surface or manifold. Its four-dimensional equivalent of a distance in three-dimensional space is called an interval. Assuming that a specific time period is represented as a real number in the same way as a distance in space, an interval in relativistic spacetime is given by the usual formula but with time negated:

where , and are distances along each spatial axis and is a period of time or "distance" along the time axis.

Mathematically this is equivalent to writing

In this context, may be either accepted as a feature of the relationship between space and real time, as above, or it may alternatively be incorporated into time itself, such that the value of is itself an imaginary number, and the equation rewritten in normalised form:

Similarly its four vector may then be written as

where distances are represented as , is the velocity of light and .

In physical cosmology, imaginary time may be incorporated into certain models of the universe which are solutions to the equations of general relativity. In particular, imaginary time can help to smooth out gravitational singularities, where known physical laws break down, to remove the singularity and avoid such breakdowns (see Hartle–Hawking state). The Big Bang, for example, appears as a singularity in ordinary time but, when modelled with imaginary time, the singularity can be removed and the Big Bang functions like any other point in four-dimensional spacetime. Any boundary to spacetime is a form of singularity, where the smooth nature of spacetime breaks down. With all such singularities removed from the Universe it thus can have no boundary and Stephen Hawking has speculated that "the boundary condition to the Universe may be that it has no boundary".

However the unproven nature of the relationship between actual physical time and imaginary time incorporated into such models has raised criticisms.[3]

In quantum statistical mechanics

It can be shown that at finite temperature , the Green's functions are periodic in imaginary time with a period of . Therefore, their Fourier transforms contain only a discrete set of frequencies called Matsubara frequencies.

Another way to see the connection between statistical mechanics and quantum field theory is to consider the transition amplitude between an initial state I and a final state F, where H is the Hamiltonian of that system. If we compare this with the partition function we see that to get the partition function from the transition amplitudes we can replace , set F = I = n and sum over n. This way we don't have to do twice the work by evaluating both the statistical properties and the transition amplitudes.

Finally by using a Wick rotation one can show that the Euclidean quantum field theory in (D + 1)-dimensional spacetime is nothing but quantum statistical mechanics in D-dimensional space.

See also

References

Notes

  1. ^ Hawking (2001), p.59.
  2. ^ Coxeter, H.S.M.; The Real Projective Plane, 3rd Edn, Springer 1993, p. 210 (footnote).
  3. ^ Robert J. Deltete & Reed A. Guy; "Emerging from Imaginary Time", Synthese, Vol. 108, No. 2 (Aug., 1996), pp. 185-203.

Bibliography

  • Stephen W. Hawking (1998). A Brief History of Time (Tenth Anniversary Commemorative ed.). Bantam Books. p. 157. ISBN 978-0-553-10953-5.
  • Hawking, Stephen (2001). The Universe in a Nutshell. United States & Canada: Bantam Books. pp. 58–61, 63, 82–85, 90–94, 99, 196. ISBN 0-553-80202-X.

Further reading

External links

A series and B series

In philosophy, A series and B series are two different descriptions of the temporal ordering relation among events. The two series differ principally in their use of tense to describe the temporal relation between events. The terms were introduced by the Scottish idealist philosopher John McTaggart in 1908 as part of his argument for the unreality of time, but since then they have become widely used terms of reference in modern discussions of the philosophy of time.

Chronometry

Chronometry (from Greek χρόνος chronos, "time" and μέτρον metron, "measure") is the science of the measurement of time, or timekeeping. Chronometry applies to electronic devices, while horology refers to mechanical devices.

It should not to be confused with chronology, the science of locating events in time, which often relies upon it.

Endurantism

Endurantism or endurance theory is a philosophical theory of persistence and identity. According to the endurantist view, material objects are persisting three-dimensional individuals wholly present at every moment of their existence, which goes with an A-theory of time. This conception of an individual as always present is opposed to perdurantism or four dimensionalism, which maintains that an object is a series of temporal parts or stages, requiring a B-theory of time. The use of "endure" and "perdure" to distinguish two ways in which an object can be thought to persist can be traced to David Lewis.

Eternity

Eternity in common parlance is an infinitely long period of time. In classical philosophy, however, eternity is defined as what exists outside time while sempiternity is the concept that corresponds to the colloquial definition of eternity.

Eternity is an important concept in many religions, where the god or gods are said to endure eternally. Some, such as Aristotle, would say the same about the natural cosmos in regard to both past and future eternal duration, and like the eternal Platonic forms, immutability was considered essential.

Event (philosophy)

In philosophy, events are objects in time or instantiations of properties in objects.

Green's function (many-body theory)

In many-body theory, the term Green's function (or Green function) is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.

The name comes from the Green's functions used to solve inhomogeneous differential equations, to which they are loosely related. (Specifically, only two-point 'Green's functions' in the case of a non-interacting system are Green's functions in the mathematical sense; the linear operator that they invert is the Hamiltonian operator, which in the non-interacting case is quadratic in the fields.)

HD2IOA

HD2IOA is the callsign of a time signal radio station operated by the Navy of Ecuador. The station is located at Guayaquil, Ecuador and transmits in the HF band on 3.81 and 7.6 MHz.The transmission is in AM mode with only the lower sideband (part of the time H3E and the rest H2B/H2D) and consists of 780 Hz tone pulses repeated every ten seconds and voice announcements in Spanish.

While sometimes this station is described as defunct, reception reports of this station on 3.81 MHz appear regularly at the Utility DX Forum.

Imaginary

Imaginary may refer to:

Imaginary (sociology), a concept in sociology

The Imaginary (psychoanalysis), a concept by Jacques Lacan

Imaginary number, a concept in mathematics

Imaginary time, a concept in physics

Imagination, a mental faculty

Object of the mind, an object of the imagination

Imaginary friend

Intercalation (timekeeping)

Intercalation or embolism in timekeeping is the insertion of a leap day, week, or month into some calendar years to make the calendar follow the seasons or moon phases. Lunisolar calendars may require intercalations of both days and months.

Minute

The minute is a unit of time or angle. As a unit of time, the minute is most of times equal to ​1⁄60 (the first sexagesimal fraction) of an hour, or 60 seconds. In the UTC time standard, a minute on rare occasions has 61 seconds, a consequence of leap seconds (there is a provision to insert a negative leap second, which would result in a 59-second minute, but this has never happened in more than 40 years under this system). As a unit of angle, the minute of arc is equal to ​1⁄60 of a degree, or 60 seconds (of arc). Although not an SI unit for either time or angle, the minute is accepted for use with SI units for both. The SI symbols for minute or minutes are min for time measurement, and the prime symbol after a number, e.g. 5′, for angle measurement. The prime is also sometimes used informally to denote minutes of time.

Multiple time dimensions

The possibility that there might be more than one dimension of time has occasionally been discussed in physics and philosophy.

Perdurantism

Perdurantism or perdurance theory is a philosophical theory of persistence and identity. The perdurantist view is that an individual has distinct temporal parts throughout its existence. Perdurantism is usually presented as the antipode to endurantism, the view that an individual is wholly present at every moment of its existence.The use of "endure" and "perdure" to distinguish two ways in which an object can be thought to persist can be traced to David Kellogg Lewis (1986). However, contemporary debate has demonstrated the difficulties in defining perdurantism (and also endurantism). For instance, the work of Ted Sider (2001) has suggested that even enduring objects can have temporal parts, and it is more accurate to define perdurantism as being the claim that objects have a temporal part at every instant that they exist. Currently there is no universally acknowledged definition of perdurantism. Others argue that this problem is avoided by creating time as a continuous function, rather than a discrete one.

Perdurantism is also referred to as "four-dimensionalism" (by Ted Sider, in particular) but perdurantism also applies if one believes there are temporal but non-spatial abstract entities (like immaterial souls or universals of the sort accepted by David Malet Armstrong).

Philosophical presentism

Philosophical presentism is the view that neither the future nor the past exist. In some versions of presentism, this view is extended to timeless objects or ideas (such as numbers). According to presentism, events and entities that are wholly past or wholly future do not exist at all. Presentism contrasts with eternalism and the growing block theory of time which hold that past events, like the Battle of Waterloo, and past entities, like Alexander the Great's warhorse Bucephalus, really do exist, although not in the present. Eternalism extends to future events as well.

Quantum superposition

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" and . Here is the Dirac notation for the quantum state that will always give the result 0 when converted to classical logic by a measurement. Likewise 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.

Reptation Monte Carlo

Reptation Monte Carlo is a quantum Monte Carlo method.

It is similar to Diffusion Monte Carlo, except that it works with paths rather than points. This has some advantages relating to calculating certain properties of the system under study that diffusion Monte Carlo has difficulty with.

In both diffusion Monte Carlo and reptation Monte Carlo, the method first aims to solve the time-dependent Schrödinger equation in the imaginary time direction. When you propagate the Schrödinger equation in time, you get the dynamics of the system under study. When you propagate it in imaginary time, you get a system that tends towards the ground state of the system.

When substituting in place of , the Schrodinger equation becomes identical with a diffusion equation. Diffusion equations can be solved by imagining a huge population of particles (sometimes called "walkers"), each diffusing in a way that solves the original equation. This is how diffusion Monte Carlo works.

Reptation Monte Carlo works in a very similar way, but is focused on the paths that the walkers take, rather than the density of walkers.

In particular, a path may be mutated using a Metropolis algorithm which tries a change (normally at one end of the path) and then accepts or rejects the change based on a probability calculation.

The update step in diffusion Monte Carlo would be moving the walkers slightly, and then duplicating and removing some of them. By contrast, the update step in reptation Monte Carlo mutates a path, and then accepts or rejects the mutation.

Thermal quantum field theory

In theoretical physics, thermal quantum field theory (thermal field theory for short) or finite temperature field theory is a set of methods to calculate expectation values of physical observables of a quantum field theory at finite temperature.

In the Matsubara formalism, the basic idea (due to Felix Bloch) is that the expectation values of operators in a canonical ensemble

may be written as expectation values in ordinary quantum field theory where the configuration is evolved by an imaginary time . One can therefore switch to a spacetime with Euclidean signature, where the above trace (Tr) leads to the requirement that all bosonic and fermionic fields be periodic and antiperiodic, respectively, with respect to the Euclidean time direction with periodicity (we are assuming natural units ). This allows one to perform calculations with the same tools as in ordinary quantum field theory, such as functional integrals and Feynman diagrams, but with compact Euclidean time. Note that the definition of normal ordering has to be altered. In momentum space, this leads to the replacement of continuous frequencies by discrete imaginary (Matsubara) frequencies and, through the de Broglie relation, to a discretized thermal energy spectrum . This has been shown to be a useful tool in studying the behavior of quantum field theories at finite temperature. It has been generalized to theories with gauge invariance and was a central tool in the study of a conjectured deconfining phase transition of Yang-Mills theory. In this Euclidean field theory, real-time observables can be retrieved by analytic continuation.

The alternative to the use of fictitious imaginary times is to use a real-time formalism which come in two forms. A path-ordered approach to real-time formalisms includes the Schwinger-Keldysh formalism and more modern variants. The latter involves replacing a straight time contour from (large negative) real initial time to by one that first runs to (large positive) real time and then suitably back to . In fact all that is needed is one section running along the real time axis as the route to the end point, , is less important. The piecewise composition of the resulting complex time contour leads to a doubling of fields and more complicated Feynman rules, but obviates the need of analytic continuations of the imaginary-time formalism. The alternative approach to real-time formalisms is an operator based approach using Bogoliubov transformations, known as thermo field dynamics. As well as Feynman diagrams and perturbation theory, other techniques such as dispersion relations and the finite temperature analog of Cutkosky rules can also be used in the real time formulation.

An alternative approach which is of interest to mathematical physics is to work with KMS states.

Universality (dynamical systems)

In statistical mechanics, universality is the observation that there are properties for a large class of systems that are independent of the dynamical details of the system. Systems display universality in a scaling limit, when a large number of interacting parts come together. The modern meaning of the term was introduced by Leo Kadanoff in the 1960s, but a simpler version of the concept was already implicit in the van der Waals equation and in the earlier Landau theory of phase transitions, which did not incorporate scaling correctly.The term is slowly gaining a broader usage in several fields of mathematics, including combinatorics and probability theory, whenever the quantitative features of a structure (such as asymptotic behaviour) can be deduced from a few global parameters appearing in the definition, without requiring knowledge of the details of the system.

The renormalization group provides an intuitively appealing, albeit mathematically non-rigorous, explanation of universality. It classifies operators in a statistical field theory into relevant and irrelevant. Relevant operators are those responsible for perturbations to the free energy, the imaginary time Lagrangian, that will affect the continuum limit, and can be seen at long distances. Irrelevant operators are those that only change the short-distance details. The collection of scale-invariant statistical theories define the universality classes, and the finite-dimensional list of coefficients of relevant operators parametrize the near-critical behavior.

Wick rotation

In physics, Wick rotation, named after Gian Carlo Wick, is a method of finding a solution to a mathematical problem in Minkowski space from a solution to a related problem in Euclidean space by means of a transformation that substitutes an imaginary-number variable for a real-number variable. This transformation is also used to find solutions to problems in quantum mechanics and other areas.

Yesterday (time)

Yesterday is a temporal construct of the relative past; literally of the day before the current day (today), or figuratively of earlier periods or times, often but not always within living memory.

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