Editing Negative mass (section)

==In quantum mechanics==
In 1928, [[Paul Dirac]]'s theory of [[elementary particle]]s, now part of the [[Standard Model]], already included negative solutions.<ref name="Dirac 1928">{{cite journal |doi=10.1098/rspa.1928.0023 |title=The Quantum Theory of the Electron |journal=Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences |volume=117 |issue=778 |pages=610–624 |year=1928 |last1=Dirac |first1=P. A. M. |bibcode=1928RSPSA.117..610D |doi-access=free }}</ref> The Standard Model is a generalization of [[quantum electrodynamics]] (QED) and negative mass is already built into the theory.

[[Mike Morris (physicist)|Morris]], [[Kip Thorne|Thorne]] and [[Ulvi Yurtsever|Yurtsever]]<ref>{{cite journal |doi=10.1103/PhysRevLett.61.1446 |pmid=10038800 |title=Wormholes, Time Machines, and the Weak Energy Condition |journal=Physical Review Letters |volume=61 |issue=13 |pages=1446–1449 |year=1988 |last1=Morris |first1=Michael S. |last2=Thorne |first2=Kip S. |last3=Yurtsever |first3=Ulvi |bibcode=1988PhRvL..61.1446M |url=https://authors.library.caltech.edu/9262/1/MORprl88.pdf }}</ref> pointed out that the quantum mechanics of the [[Casimir effect]] can be used to produce a locally energy-negative region of space–time. In this article, and subsequent work by others, they showed that negative matter could be used to stabilize a [[wormhole]]. [[John G. Cramer|Cramer]] ''et al.'' argue that such wormholes might have been created in the early universe, stabilized by negative-mass loops of [[cosmic string]].<ref>{{cite journal |arxiv=astro-ph/9409051 |doi=10.1103/PhysRevD.51.3117 |title=Natural wormholes as gravitational lenses |journal=Physical Review D |volume=51 |issue=6 |pages=3117–3120 |year=1995 |last1=Cramer |first1=John G. |last2=Forward |first2=Robert L. |last3=Morris |first3=Michael S. |last4=Visser |first4=Matt |last5=Benford |first5=Gregory |last6=Landis |first6=Geoffrey A. |bibcode=1995PhRvD..51.3117C |pmid=10018782 |s2cid=42837620 }}</ref> [[Stephen Hawking]] has argued that [[negative energy]] is a necessary condition for the creation of a [[closed timelike curve]] by manipulation of gravitational fields within a finite region of space;<ref name="futureofspacetime">{{Cite book | last = Hawking | first = Stephen | title = The Future of Spacetime | publisher = W. W. Norton | year = 2002 | pages = [https://archive.org/details/futureofspacetim0000unse/page/96 96] | isbn = 978-0-393-02022-9 | url-access = registration | url = https://archive.org/details/futureofspacetim0000unse/page/96 }}</ref> this implies, for example, that a finite [[Tipler cylinder]] cannot be used as a [[Time travel|time machine]].

===Schrödinger equation===
For energy eigenstates of the [[Schrödinger equation]], the wavefunction is wavelike wherever the particle's energy is greater than the local potential, and exponential-like (evanescent) wherever it is less. Naively, this would imply kinetic energy is negative in evanescent regions (to cancel the local potential). However, kinetic energy is an operator in [[quantum mechanics]], and its expectation value is always positive, summing with the expectation value of the potential energy to yield the energy eigenvalue.

For wavefunctions of particles with zero rest mass (such as [[photon]]s), this means that any evanescent portions of the wavefunction would be associated with a local negative mass–energy. However, the Schrödinger equation does not apply to massless particles; instead the [[Klein–Gordon equation]] is required.