Quantum geometry

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v t e

In

has a profound effect on physical phenomena.

Quantum gravity

Each theory of

wrapped on this cycle.

In an alternative approach to quantum gravity called

It is possible (but considered unlikely) that this strictly quantized understanding of geometry will be consistent with the quantum picture of geometry arising from string theory.

Another, quite successful, approach, which tries to reconstruct the geometry of space-time from "first principles" is

Discrete Lorentzian quantum gravity

.

Quantum states as differential forms

Main article:

Wavefunction

See also:

Differential forms

${\displaystyle |\psi \rangle =\int \psi (\mathbf {x} ,t)\,|\mathbf {x} ,t\rangle \,\mathrm {d} ^{3}\mathbf {x} }$

where the

position vector

is

${\displaystyle \mathbf {x} =(x^{1},x^{2},x^{3})}$

the differential volume element is

${\displaystyle \mathrm {d} ^{3}\mathbf {x} =\mathrm {d} x^{1}\!\wedge \mathrm {d} x^{2}\!\wedge \mathrm {d} x^{3}}$

and x¹, x², x³ are an arbitrary set of coordinates, the upper indices indicate contravariance, lower indices indicate covariance, so explicitly the quantum state in differential form is:

${\displaystyle |\psi \rangle =\int \psi (x^{1},x^{2},x^{3},t)\,|x^{1},x^{2},x^{3},t\rangle \,\mathrm {d} x^{1}\!\wedge \mathrm {d} x^{2}\!\wedge \mathrm {d} x^{3}}$

The overlap integral is given by:

${\displaystyle \langle \chi |\psi \rangle =\int \chi ^{*}\psi ~\mathrm {d} ^{3}\mathbf {x} }$

in differential form this is

${\displaystyle \langle \chi |\psi \rangle =\int \chi ^{*}\psi ~\mathrm {d} x^{1}\!\wedge \mathrm {d} x^{2}\!\wedge \mathrm {d} x^{3}}$

The probability of finding the particle in some region of space R is given by the integral over that region:

${\displaystyle \langle \psi |\psi \rangle =\int _{R}\psi ^{*}\psi ~\mathrm {d} x^{1}\!\wedge \mathrm {d} x^{2}\!\wedge \mathrm {d} x^{3}}$

provided the wave function is normalized. When R is all of 3d position space, the integral must be 1 if the particle exists.

Differential forms are an approach for describing the geometry of

. For generality, a formalism which can be used in any coordinate system is useful.

See also

• Noncommutative geometry

References

Further reading

• Supersymmetry, Demystified, P. Labelle, McGraw-Hill (USA), 2010,
  ISBN 978-0-07-163641-4
• Quantum Mechanics, E. Abers, Pearson Ed., Addison Wesley, Prentice Hall Inc, 2004,
  ISBN 9780131461000
• Quantum Mechanics Demystified, D. McMahon, Mc Graw Hill (USA), 2006,
  ISBN 0-07-145546 9
• Quantum Field Theory, D. McMahon, Mc Graw Hill (USA), 2008,
  ISBN 978-0-07-154382-8

External links

• Space and Time: From Antiquity to Einstein and Beyond
• Quantum Geometry and its Applications