You do not have permission to edit this page, for the following reasons:

This IP address has been blocked from editing Wikipedia.
This does not affect your ability to read Wikipedia pages.
Most people who see this message have done nothing wrong. Some kinds of blocks restrict editing from specific service providers or telecom companies in response to recent abuse or vandalism, and can sometimes affect other users who are unrelated to that abuse. Review the information below for assistance if you do not believe that you have done anything wrong.

The IP address or range 35.170.0.0/16 has been blocked (disabled) by ‪JJMC89‬ for the following reason(s):

The IP address that you are currently using has been blocked because it is believed to be a web host provider or colocation provider. To prevent abuse, web hosts and colocation providers may be blocked from editing Wikipedia.
You will not be able to edit Wikipedia using a web host or colocation provider because it hides your IP address, much like a proxy or VPN.
We recommend that you attempt to use another connection to edit. For example, if you use a proxy or VPN to connect to the internet, turn it off when editing Wikipedia. If you edit using a mobile connection, try using a Wi-Fi connection, and vice versa. If you are using a corporate internet connection, switch to a different Wi-Fi network. If you have a Wikipedia account, please log in.
If you do not have any other way to edit Wikipedia, you will need to request an IP block exemption.

How to appeal if you are confident that your connection does not use a colocation provider's IP address:
If you are confident that you are not using a web host, you may appeal this block by adding the following text on your talk page: ((unblock|reason=Caught by a colocation web host block but this host or IP is not a web host. My IP address is _______. Place any further information here. ~~~~)). You must fill in the blank with your IP address for this block to be investigated. Your IP address can be determined here. Alternatively, if you wish to keep your IP address private you can use the unblock ticket request system. There are several reasons you might be editing using the IP address of a web host or colocation provider (such as if you are using VPN software or a business network); please use this method of appeal only if you think your IP address is in fact not a web host or colocation provider.

Administrators: The IP block exemption user right should only be applied to allow users to edit using web host in exceptional circumstances, and requests should usually be directed to the functionaries team via email. If you intend to give the IPBE user right, a CheckUser needs to take a look at the account. This can be requested most easily at SPI Quick Checkuser Requests. Unblocking an IP or IP range with this template is highly discouraged without at least contacting the blocking administrator.

This block will expire on 05:20, 2 July 2025.

Even when blocked, you will usually still be able to edit your user talk page, as well as email administrators and other editors.

For information on how to proceed, please read the FAQ for blocked users and the guideline on block appeals. The guide to appealing blocks may also be helpful.

Other useful links: Blocking policy · Help:I have been blocked
Your IP address is in a range that has been blocked on all Wikimedia Foundation wikis. The block was made by ‪Jon Kolbert‬. The reason given is Open proxy/Webhost: See the help page if you are affected .
- Start of block: 05:38, 21 February 2024
- Expiry of block: 05:38, 21 February 2029
Your current IP address is 35.170.198.190. The blocked range is 35.170.0.0/16. Please include all above details in any queries you make. If you believe you were blocked by mistake, you can find additional information and instructions in the No open proxies global policy. Otherwise, to discuss the block please post a request for review on Meta-Wiki. You could also send an email to the stewards VRT queue at stewards@wikimedia.org including all above details.

You can view and copy the source of this page:

((Short description|Concept in special relativity))
'''Imaginary time''' is a mathematical representation of time that appears in some approaches to [[special relativity]] and [[quantum mechanics]]. It finds uses in connecting quantum mechanics with [[statistical mechanics]] and in certain [[cosmology|cosmological]] theories.((cn|date=January 2023|reason=The section on quantum statistical mechanics does not mention imaginary time))

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

==Origins==
In mathematics, the imaginary unit <math>i</math> is the square root of <math>-1</math>, such that <math>i^2</math> is defined to be <math>-1</math>. A number which is a direct multiple of <math>i</math> is known as an [[imaginary number]].<ref name="Penrose2004">
((cite book |last=Penrose |first=Roger |author-link=Roger_Penrose |date=2004 |title=The Road to Reality |url=https://books.google.com/books?id=csaaQgAACAAJ |location= |publisher=[[Jonathan Cape]] |page= |isbn=9780224044479))</ref>((rp|Chp 4))

In certain physical theories, periods of time are multiplied by <math>i</math> in this way. Mathematically, an imaginary time period <math display="inline">\tau</math> may be obtained from real time <math display="inline"> t</math> via a [[Wick rotation]] by <math display="inline">\pi/2</math> in the [[complex plane]]: <math display="inline">\tau = it</math>.<ref name="Penrose2004" />((rp|769))

[[Stephen Hawking]] popularized the concept of imaginary time in his book ''[[The Universe in a Nutshell]]''.
((quote|"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 [[Positivism|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?"||[[Stephen Hawking]]<ref name="Hawking2001">((cite book |last=Hawking |first=Stephen W. |author-link=Stephen Hawking |title=The Universe in a Nutshell |publisher=[[Bantam Books]] |date=November 2001 |location=United States & Canada |pages=58–61, 63, 82–85, 90–94, 99, 196 |isbn=9780553802023 |ol=7850510M |url=https://openlibrary.org/books/OL7850510M/The_Universe_in_a_Nutshell#bookPreview))</ref>((rp|59))))

In fact, the terms "[[real number|real]]" and "[[imaginary number|imaginary]]" for numbers are just a historical accident, much like the terms "[[rational number|rational]]" and "[[irrational number|irrational]]":
((quote|"...the words ''real'' and ''imaginary'' are picturesque relics of an age when the nature of [[complex number]]s was not properly understood."||[[Harold Scott MacDonald Coxeter|H.S.M. Coxeter]]<ref>((cite book |last=Coxeter |first=H.S.M. |author-link=Harold Scott MacDonald Coxeter |date=1949 |title=The Real Projective Plane |url=https://archive.org/details/in.ernet.dli.2015.212962/page/n197 |location=New York |publisher=[[McGraw-Hill Book Company]] |page=187 footnote))</ref>))

== In cosmology ==
===Derivation===
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 [[spacetime interval|interval]]. Assuming that a specific time period is represented as a [[real number]] in the same way as a distance in space, an interval <math>d</math> in relativistic spacetime is given by the usual formula but with time negated:
<math display="block">d^2 = x^2 + y^2 + z^2 - t^2</math>
where <math>x</math>, <math>y</math> and <math>z</math> are distances along each spatial axis and <math>t</math> is a period of time or "distance" along the time axis (Strictly, the time coordinate is <math>(ct)^2</math> where <math>c</math> is the [[speed of light]], however we conventionally choose units such that <math>c=1</math>).

Mathematically this is equivalent to writing
<math display="block">d^2 = x^2 + y^2 + z^2 + (it)^2</math>

In this context, <math>i</math> 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]], denoted by <math>\tau</math>.((cn|date=January 2023|reason=this is the critical statement)) The equation may then be rewritten in normalised form: <math display="block">d^2 = x^2 + y^2 + z^2 + \tau^2</math>

Similarly its [[four vector]] may then be written as <math display="block">( x_0, x_1, x_2, x_3 )</math> where distances are represented as <math>x_n</math>, and <math>x_0 = ict</math> where <math>c</math> is the speed of light and time is imaginary.

===Application to cosmology===
Hawking noted the utility of rotating time intervals into an imaginary metric in certain situations, in 1971.<ref>((cite journal |last=Hawking |first=S. W. |author-link=Stephen_Hawking |title=Quantum gravity and path integrals |journal=[[Phys. Rev. D]] |volume=18 |issue=6 |date=1978-09-15 |pages=1747–1753 |doi=10.1103/PhysRevD.18.1747 |bibcode=1978PhRvD..18.1747H |url=https://journals.aps.org/prd/abstract/10.1103/PhysRevD.18.1747 |access-date=2023-01-25 |url-access=subscription|quote=It is convenient to rotate the time interval on this timelike tube between the two surfaces into the complex plane so that it becomes purely imaginary.))</ref>

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 [[Gravitational singularity|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.<ref name="Penrose2004" />((rp|769-772)) With all such singularities removed from the Universe, it thus can have no boundary and Stephen Hawking speculated that "the [[boundary condition]] to the Universe is that it has no boundary".<ref name="Hawking2001" />((rp|85))

However, the unproven nature of the relationship between actual physical time and imaginary time incorporated into such models has raised criticisms.<ref>((cite journal |last1=Deltete |first1=Robert J.|last2=Guy |first2=Reed A. |title=Emerging from imaginary time |journal=[[Synthese]] |date=Aug 1996 |volume=108 |issue=2 |pages=185–203 |doi=10.1007/BF00413497|s2cid=44131608 |url=https://link.springer.com/article/10.1007/BF00413497 |access-date=2023-01-25 |url-access=subscription))</ref> [[Roger Penrose]] has noted that there needs to be a transition from the [[Riemannian_manifold#Riemannian_metrics|Riemannian metric]] (often referred to as "[[Euclidean_metric|Euclidean]]" in this context) with imaginary time at the Big Bang to a [[Pseudo-Riemannian_manifold#Lorentzian_manifold|Lorentzian 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.<ref name="Penrose2004" />((rp|769-772))

== In quantum statistical mechanics ==
((refimprove|section|date=November 2017))

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 [[quantum harmonic oscillators|harmonic oscillators]] of [[quantum field theory]].<ref>
((cite web |url=http://www.wiese.itp.unibe.ch/lectures/fieldtheory.pdf |title=Quantum Field Theory |last=Wiese |first=Uwe-Jens |date=2007-08-21 |website=Institute for Theoretical Physics |publisher=University of Bern |access-date=2023-01-25 |page=63))</ref> The [[Green's function]] of an inhomogeneous linear [[differential operator]], defined on a domain with specified initial conditions or [[boundary conditions]], is its [[Impulse_(physics)|impulse]] response, and mathematically we define the point particles of statistical mechanics as [[Dirac delta functions]], which is to say impulses. At a finite temperature <math>T</math>, the [[Green's function]]s are [[periodic function|periodic]] in imaginary time with a period of <math display="inline"> 2\beta = 2/T</math>. Therefore, their Fourier transforms contain only a discrete set of frequencies called [[Matsubara frequency|Matsubara frequencies]]. 

The connection between statistical mechanics and quantum field theory is also seen in the [[transition amplitude]] <math display="inline">\langle F\mid e^{-itH}\mid I\rangle </math> between an initial state ((math|''I'')) and a final state&nbsp;((math|''F'')), where ((math|''H''))&nbsp;is the [[Hamiltonian (quantum mechanics)|Hamiltonian]] of that system. Comparing this with the [[partition function (quantum field theory)|partition function]] <math display="inline"> Z = \operatorname{Tr} e^{-\beta H}</math> shows that the partition function may be derived from the transition amplitudes by substituting <math display="inline"> t = \beta/i</math>, setting ((math|1=''F'' = ''I'' = ''n'')) and summing over ((math|''n'')). 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 (''D''&nbsp;+&nbsp;1)-dimensional spacetime is nothing but [[quantum statistical mechanics]] in ''D''-dimensional space.

==See also==
*[[Euclidean quantum gravity]]
*[[Multiple time dimensions]]

== References ==
((reflist))

== Further reading ==
*((cite book|author=Hawking|title=A Brief History of Time|first=Stephen W.|url=https://books.google.com/books?id=E7mEnx3zE4AC|edition=Tenth Anniversary Commemorative|year=1998|publisher=Bantam Books|language=en|isbn=978-0-553-10953-5|location=|page=157))
* [https://www.amazon.com/dp/0306463385/ Gerald D. Mahan. Many-Particle Physics, Chapter 3]
* [https://www.amazon.com/dp/0691010196/ A. Zee Quantum field theory in a nutshell, Chapter V.2]

== External links ==
* [https://www.hawking.org.uk/in-words/lectures/the-beginning-of-time The Beginning of Time] — Lecture by Stephen Hawking which discusses imaginary time.
* [https://www.pbs.org/wnet/hawking/strange/html/imaginary.html Stephen Hawking's Universe: Strange Stuff Explained] ((Webarchive|url=https://web.archive.org/web/20160303174253/http://www.pbs.org/wnet/hawking/strange/html/imaginary.html |date=2016-03-03 )) — PBS site on imaginary time.

[[Category:Quantum mechanics]]
[[Category:Philosophy of time]]

Pages transcluded onto the current version of this page (help):

Imaginary time (edit)
Template:Ambox (view source) (template editor protected)
Template:Blockquote (view source) (template editor protected)
Template:Blockquote/styles.css (view source) (template editor protected)
Template:Category handler (view source) (protected)
Template:Citation needed (view source) (protected)
Template:Cite book (view source) (protected)
Template:Cite journal (view source) (protected)
Template:Cite web (view source) (protected)
Template:Cn (view source) (template editor protected)
Template:Comma separated entries (view source) (template editor protected)
Template:Delink (view source) (protected)
Template:Find sources mainspace (view source) (template editor protected)
Template:Fix (view source) (protected)
Template:Fix/category (view source) (protected)
Template:If empty (view source) (protected)
Template:Main other (view source) (protected)
Template:Math (view source) (template editor protected)
Template:More citations needed (view source) (template editor protected)
Template:Pagetype (view source) (protected)
Template:Quote (view source) (template editor protected)
Template:R/superscript (view source) (template editor protected)
Template:R/where (view source) (template editor protected)
Template:Refimprove (view source) (template editor protected)
Template:Reflist (view source) (protected)
Template:Reflist/styles.css (view source) (protected)
Template:Rp (view source) (template editor protected)
Template:SDcat (view source) (protected)
Template:Short description (view source) (protected)
Template:Short description/lowercasecheck (view source) (protected)
Template:Sp (view source) (template editor protected)
Template:Webarchive (view source) (template editor protected)
Module:Arguments (view source) (protected)
Module:Category handler (view source) (protected)
Module:Category handler/blacklist (view source) (protected)
Module:Category handler/config (view source) (protected)
Module:Category handler/data (view source) (protected)
Module:Category handler/shared (view source) (protected)
Module:Check for unknown parameters (view source) (protected)
Module:Citation/CS1 (view source) (protected)
Module:Citation/CS1/COinS (view source) (protected)
Module:Citation/CS1/Configuration (view source) (protected)
Module:Citation/CS1/Date validation (view source) (protected)
Module:Citation/CS1/Identifiers (view source) (protected)
Module:Citation/CS1/Utilities (view source) (protected)
Module:Citation/CS1/Whitelist (view source) (protected)
Module:Citation/CS1/styles.css (view source) (protected)
Module:DecodeEncode (view source) (template editor protected)
Module:Delink (view source) (protected)
Module:Disambiguation/templates (view source) (protected)
Module:Find sources (view source) (template editor protected)
Module:Find sources/config (view source) (template editor protected)
Module:Find sources/links (view source) (template editor protected)
Module:Find sources/templates/Find sources mainspace (view source) (template editor protected)
Module:GetParameters (view source) (template editor protected)
Module:If empty (view source) (protected)
Module:Message box (view source) (protected)
Module:Message box/ambox.css (view source) (protected)
Module:Message box/configuration (view source) (protected)
Module:Namespace detect/config (view source) (protected)
Module:Namespace detect/data (view source) (protected)
Module:Pagetype (view source) (protected)
Module:Pagetype/config (view source) (protected)
Module:Pagetype/disambiguation (view source) (protected)
Module:Pagetype/setindex (view source) (protected)
Module:Pagetype/softredirect (view source) (protected)
Module:Plain text (view source) (template editor protected)
Module:SDcat (view source) (protected)
Module:Separated entries (view source) (protected)
Module:String (view source) (protected)
Module:String2 (view source) (template editor protected)
Module:TableTools (view source) (protected)
Module:Unsubst (view source) (protected)
Module:Webarchive (view source) (template editor protected)
Module:Webarchive/data (view source) (template editor protected)
Module:Wikitext Parsing (view source) (protected)
Module:Yesno (view source) (protected)

Return to Imaginary time.

Retrieved from "https://en.wikipedia.org/wiki/Imaginary_time"