- Hypothetical particles which have "exotic" physical properties that would violate known laws of physics, such as a particle having a negative mass.
- Hypothetical particles which have not yet been encountered, such as exotic baryons, but whose properties would be within the realm of mainstream physics if found to exist. Futurist Ray Kurzweil has speculated that by the end of the 21st century it may be possible by using femtotechnology to create new chemical elements composed of exotic baryons that would eventually constitute a new periodic table of elements in which the elements would have completely different properties from the regular chemical elements.
- States of matter which are not commonly encountered, such as Bose–Einstein condensates and quark–gluon plasma, but whose properties are perfectly within the realm of mainstream physics.
- States of matter which are poorly understood, such as dark matter.
Negative mass would possess some strange properties, such as accelerating in the direction opposite of applied force. For example, an object with negative inertial mass and positive electric charge would accelerate away from objects with negative charge, and towards objects with positive charge, the opposite of the normal rule that like charges repel and opposite charges attract. This behaviour can produce bizarre results: for instance, a gas containing a mixture of positive and negative matter particles will have the positive matter portion increase in temperature without bound. However, the negative matter portion gains negative temperature at the same rate, again balancing out.
Despite being inconsistent with the expected behavior of "normal" matter, negative mass is mathematically consistent and introduces no violation of conservation of momentum or energy. It is used in certain speculative theories, such as on the construction of wormholes. The closest known real representative of such exotic matter is the region of pseudo-negative-pressure density produced by the Casimir effect.
If the rest mass is imaginary this implies that the denominator is imaginary because the total energy is an observable and thus must be real; therefore the quantity under the square root must be negative, which can only happen if v is greater than c. As noted by Gregory Benford et al., special relativity implies that tachyons, if they existed, could be used to communicate backwards in time (see Tachyonic antitelephone). Because time travel is considered to be non-physical, tachyons are believed by physicists either to not exist, or else to be incapable of interacting with normal matter.
In quantum field theory, imaginary mass would induce tachyon condensation.