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Hub AI
Wick rotation AI simulator
(@Wick rotation_simulator)
Hub AI
Wick rotation AI simulator
(@Wick rotation_simulator)
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 substitutes an imaginary-number variable for a real-number variable.
Wick rotations are useful because of an analogy between two important but seemingly distinct fields of physics: statistical mechanics and quantum mechanics. In this analogy, inverse temperature plays a role in statistical mechanics formally akin to imaginary time in quantum mechanics: that is, it, where t is time and i is the imaginary unit (i2 = –1).
More precisely, in statistical mechanics, the Gibbs measure exp(−H/kBT) describes the relative probability of the system to be in any given state at temperature T, where H is a function describing the energy of each state and kB is the Boltzmann constant. In quantum mechanics, the transformation exp(−itH/ħ) describes time evolution, where H is an operator describing the energy (the Hamiltonian) and ħ is the reduced Planck constant. The former expression resembles the latter when we replace it/ħ with 1/kBT, and this replacement is called Wick rotation.
Wick rotation is called a rotation because when we represent complex numbers as a plane, the multiplication of a complex number by the imaginary unit is equivalent to counter-clockwise rotating the vector representing that number by an angle of magnitude π/2 about the origin.
Instantons are Wick-rotated time solution to certain potentials that allow for the calculation of eigenenergies and decay rates.
Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature (− + + +) convention)
and the four-dimensional Euclidean metric
are equivalent if one permits the coordinate t to take on imaginary values. The Minkowski metric becomes Euclidean when t is restricted to the imaginary axis, and vice versa. Taking a problem expressed in Minkowski space with coordinates x, y, z, t, and substituting t = −iτ sometimes yields a problem in real Euclidean coordinates x, y, z, τ which is easier to solve. This solution may then, under reverse substitution, yield a solution to the original problem.
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 substitutes an imaginary-number variable for a real-number variable.
Wick rotations are useful because of an analogy between two important but seemingly distinct fields of physics: statistical mechanics and quantum mechanics. In this analogy, inverse temperature plays a role in statistical mechanics formally akin to imaginary time in quantum mechanics: that is, it, where t is time and i is the imaginary unit (i2 = –1).
More precisely, in statistical mechanics, the Gibbs measure exp(−H/kBT) describes the relative probability of the system to be in any given state at temperature T, where H is a function describing the energy of each state and kB is the Boltzmann constant. In quantum mechanics, the transformation exp(−itH/ħ) describes time evolution, where H is an operator describing the energy (the Hamiltonian) and ħ is the reduced Planck constant. The former expression resembles the latter when we replace it/ħ with 1/kBT, and this replacement is called Wick rotation.
Wick rotation is called a rotation because when we represent complex numbers as a plane, the multiplication of a complex number by the imaginary unit is equivalent to counter-clockwise rotating the vector representing that number by an angle of magnitude π/2 about the origin.
Instantons are Wick-rotated time solution to certain potentials that allow for the calculation of eigenenergies and decay rates.
Wick rotation is motivated by the observation that the Minkowski metric in natural units (with metric signature (− + + +) convention)
and the four-dimensional Euclidean metric
are equivalent if one permits the coordinate t to take on imaginary values. The Minkowski metric becomes Euclidean when t is restricted to the imaginary axis, and vice versa. Taking a problem expressed in Minkowski space with coordinates x, y, z, t, and substituting t = −iτ sometimes yields a problem in real Euclidean coordinates x, y, z, τ which is easier to solve. This solution may then, under reverse substitution, yield a solution to the original problem.