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Imaginary time is a mathematical representation of time which appears in some approaches to
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
and
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. It finds uses in connecting quantum mechanics with
statistical mechanics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic b ...
and in certain
cosmological Cosmology () is a branch of physics and metaphysics dealing with the nature of the universe. The term ''cosmology'' was first used in English in 1656 in Thomas Blount's ''Glossographia'', and in 1731 taken up in Latin by German philosopher ...
theories. Mathematically, imaginary time is real time which has undergone a
Wick rotation In physics, Wick rotation, named after Italian physicist 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 s ...
so that its coordinates are multiplied by the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
''i''. Imaginary time is ''not'' imaginary in the sense that it is unreal or made-up (any more than, say,
irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...
s defy logic), it is simply expressed in terms of what mathematicians call
imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . F ...
s.


Origins

In mathematics, the imaginary unit i is the square root of -1, such that i^2 is defined to be -1. A number which is a direct multiple of i is known as an
imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . F ...
. In certain physical theories, periods of time are multiplied by i in this way. Mathematically, an imaginary time period \tau may be obtained from real time t via a
Wick rotation In physics, Wick rotation, named after Italian physicist 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 s ...
by \pi/2 in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
: \tau = it. Stephen Hawking popularized the concept of imaginary time in his book ''
The Universe in a Nutshell ''The Universe in a Nutshell'' is a 2001 book about theoretical physics by Stephen Hawking. It is generally considered a sequel and was created to update the public concerning developments since the multi-million-copy bestseller ''A Brief Histor ...
''. In fact, the terms "real" and "imaginary" for numbers are just a historical accident, much like the terms "
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abi ...
" and "
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
":


In cosmology

In the
Minkowski spacetime In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the ...
model adopted by the
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
, 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 mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
in the same way as a distance in space, an interval d in relativistic spacetime is given by the usual formula but with time negated: d^2 = x^2 + y^2 + z^2 - t^2 where x, y and z are distances along each spatial axis and t is a period of time or "distance" along the time axis (Strictly, the time coordinate is (ct)^2 where c is the speed of light, however we conventionally choose units such that c=1). Mathematically this is equivalent to writing d^2 = x^2 + y^2 + z^2 + (it)^2 In this context, i 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 time is itself an
imaginary number An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . F ...
, denoted by \tau, and the equation rewritten in normalised form: d^2 = x^2 + y^2 + z^2 + \tau^2 Similarly its
four vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
may then be written as ( x_0, x_1, x_2, x_3 ) where distances are represented as x_n, c is the
velocity of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special relativity, special theory of relativity, is ...
and x_0 = ict. Hawking noted the utility of rotating time intervals into an imaginary metric in certain situations, in 1971. In
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
, imaginary time may be incorporated into certain models of the
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
which are solutions to the equations of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
. In particular, imaginary time can help to smooth out
gravitational singularities A gravitational singularity, spacetime singularity or simply singularity is a condition in which gravity is so intense that spacetime itself breaks down catastrophically. As such, a singularity is by definition no longer part of the regular ...
, where known physical laws break down, to remove the singularity and avoid such breakdowns (see
Hartle–Hawking state The Hartle–Hawking state is a proposal in theoretical physics concerning the state of the Universe prior to the Planck epoch. It is named after James Hartle and Stephen Hawking. Hartle and Hawking suggest that if we could travel backwards in t ...
). The
Big Bang The Big Bang event is a physical theory that describes how the universe expanded from an initial state of high density and temperature. Various cosmological models of the Big Bang explain the evolution of the observable universe from the ...
, 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 In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differ ...
. Any boundary to spacetime is a form of singularity, where the smooth nature of spacetime breaks down.Penrose (2004). pp.769-772. With all such singularities removed from the Universe, it thus can have no boundary and Stephen Hawking speculated that "the
boundary condition In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to th ...
to the Universe is 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.
Roger Penrose Sir Roger Penrose (born 8 August 1931) is an English mathematician, mathematical physicist, philosopher of science and Nobel Laureate in Physics. He is Emeritus Rouse Ball Professor of Mathematics in the University of Oxford, an emeritus f ...
has noted that there needs to be a transition from the Riemannian metric (often referred to as "Euclidean" in this context) with imaginary time at the Big Bang to a Lorenzian metric with real time for the evolving Universe. Also, modern observations suggest that the Universe is open and will never shrink back to a Big Crunch. If this proves true, then the end-of-time boundary still remains.


In quantum statistical mechanics

The equations of the quantum field can be obtained by taking the Fourier transform of the equations of statistical mechanics. Since the Fourier transform of a function typically shows up as its inverse, the point particles of statistical mechanics become, under a Fourier transform, the infinitely extended harmonic oscillators of quantum field theory.Uwe-Jens Wiese
"Quantum Field Theory"
Institute for Theoretical Physics, University of Bern, 21 August 2007, page 63.
The
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
of an inhomogeneous linear differential operator, defined on a domain with specified initial conditions or boundary conditions, is its impulse response, and mathematically we define the point particles of statistical mechanics as Dirac delta functions, which is to say impulses. At finite temperature T, the
Green's function In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if \operatorname is the linear differenti ...
s are periodic in imaginary time with a period of 2\beta = 2/T. Therefore, their
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
s contain only a discrete set of frequencies called Matsubara frequencies. The connection between statistical mechanics and quantum field theory is also seen in the transition amplitude \langle F\mid e^\mid I\rangle between an initial state and a final state , where  is the Hamiltonian of that system. Comparing this with the partition function Z = \operatorname e^ shows that the partition function may be derived from the transition amplitudes by substituting t = \beta/i, setting and summing over . This avoids the need 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 theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles and ...
in (''D'' + 1)-dimensional spacetime is nothing but
quantum statistical mechanics Quantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics a statistical ensemble (probability distribution over possible quantum states) is described by a density operator ''S'', which is ...
in ''D''-dimensional space.


See also

*
Euclidean quantum gravity In theoretical physics, Euclidean quantum gravity is a version of quantum gravity. It seeks to use the Wick rotation to describe the force of gravity according to the principles of quantum mechanics. Introduction in layperson's terms The W ...
*
Multiple time dimensions The possibility that there might be more than one dimension of time has occasionally been discussed in physics and philosophy. Similar ideas appear in folklore and fantasy literature. Physics Speculative theories with more than one time dimen ...


References


Notes


Bibliography

* *{{cite book, last=Hawking, first=Stephen W., title=The Universe in a Nutshell, publisher=Bantam Books, date=2001, location=United States & Canada, pages=58–61, 63, 82–85, 90–94, 99, 196, isbn=0-553-80202-X * Penrose, Roger (2004). ''The Road to Reality'', Jonathan Cape. (softback, Vintage Books, 2005).


Further reading


Gerald D. Mahan. Many-Particle Physics, Chapter 3

A. Zee Quantum field theory in a nutshell, Chapter V.2


External links


The Beginning of Time
— Lecture by Stephen Hawking which discusses imaginary time.

— PBS site on imaginary time. Quantum mechanics Philosophy of time