Поворот Вика

В физике, Поворот Вика, названный в честь Жана-Карла Вика, это метод решения задачи в пространстве Минковского с помощью решения связанной проблемы в Евклидовом пространстве, используя комплексный анализ, в частности аналитическое продолжение.

Обзор

Поворот Вика мотивирован наблюдением, что метрический тензор Минковского (с сигнатурой (-1,1,1,1))

$ds^{2}=-(dt^{2})+dx^{2}+dy^{2}+dz^{2}$

и четырёхмерный евклидов метрический тензор

$ds^{2}=d\tau ^{2}+dx^{2}+dy^{2}+dz^{2}$

есть эквивалентными, если разрешить координате t принимать комплексные значение. Метрика Минковского стает Евклидовой, если разрешить t принимать только мнимые значения, тоже самое и для евклидовой метрики. Таким способом иногда можно задачу в пространстве Минковского с координатами x, y, z, t, заменяя $t=i\tau$, свести к задаче в действительном евклидовом пространстве с координатами x, y, z, $\tau$ которую легче решить. Это решение можно затем, после обратной замены, свести к начальной задаче.

Статистическая и квантовая механика

Поворот Вика связывает статистическую механику с квантовой с помощью замены обратной температуры $1/(k_{B}T)\,$ мнимым временем $it/\hbar \,$. Рассмотрим большое число гармонических осцилляторов при температуре $T\,$. Относительная вероятность найти заданный осциллятор в состоянии с энергией $E\,$ is $\exp(-E/k_{B}T)\,$, гду $k_{B}\,$ константа Больцмана. Среднее значение наблюдаемой $Q\,$ это, без нормирующего множителя,

$\sum _{j}Q(j)e^{-E_{j}/(k_{B}T)}.\,$

Now consider a single quantum harmonic oscillator in a superposition of basis states, evolving for a time $t$ under a Hamiltonian $H$. The relative phase change of the basis state with energy $E\,$ is $\exp(-Eit/\hbar ),\,$ where $\hbar \,$ is Planck's constant. The probability amplitude that a uniform superposition of states $|\psi \rangle =\sum _{j}|j\rangle \,$evolves to an arbitrary superposition $|Q\rangle =\sum _{j}Q_{j}|j\rangle \,$ is, up to a normalizing constant,

$\;\langle Q|e^{-iHt/\hbar }|\psi \rangle$

$=\sum _{j}Q_{j}e^{-E_{j}it/\hbar }\langle j|j\rangle$

$=\sum _{j}Q_{j}e^{-E_{j}it/\hbar }.$

Statics and dynamics

Wick rotation relates statics problems in $n$ dimensions to dynamics problems in $n-1$ dimensions, trading one dimension of space for one dimension of time. A simple example where $n=2$ is a hanging spring with fixed endpoints in a gravitational field. The shape of the spring is a curve $y(x)$. The spring is in equilibrium when the energy associated with this curve is at a critical point; this critical point is typically a minimum, so this idea is usually called "the principle of least energy". To compute the energy, we integrate over the energy density at each point:

$E=\int _{x}\left[k\left({\frac {dy(x)}{dx}}\right)^{2}+V(x)\right]dx,$

where $k$ is the spring constant and $V(x)$ is the gravitational potential.

The corresponding dynamics problem is that of a rock thrown upwards; the path the rock follows is a critical point of the action. Action is the integral of the Lagrangian; as before, this critical point is typically a minimum, so this is called the "principle of least action":

$S=\int _{t}\left[m\left({\frac {dy(t)}{dt}}\right)^{2}-V(t)\right]dt$

We get the solution to the dynamics problem (up to a factor of $-i$) from the statics problem by Wick rotation, replacing $x$ by $t$, $dx$ by $idt$, and the spring constant $k$ by the mass of the rock $m$:

$-iS=\int _{t}\left[m\left({\frac {dy(t)}{idt}}\right)^{2}+V(t)\right](idt)$

$=-i\int _{t}\left[m\left({\frac {dy(t)}{dt}}\right)^{2}-V(t)\right]dt$

Both thermal/quantum and static/dynamic

Taken together, the previous two examples show how the path integral formulation of quantum mechanics is related to statistical mechanics: the shape of each spring in a collection at temperature $T$ will deviate from the least-energy shape due to thermal fluctuations; the probability of finding a spring with a given shape decreases exponentially with the energy difference from the least-energy shape. Similarly, a quantum particle moving in a potential can be described by a superposition of paths, each with a phase $\exp(-iS)$: the thermal variations in the shape across the collection have turned into quantum uncertainty in the path of the quantum particle.

Others

The Schrödinger equation and the heat equation are also related by Wick rotation. However, there is a slight difference. Statistical mechanics n-point functions satisfy positivity whereas Wick-rotated quantum field theories satisfy reflection positivity.

It is called a rotation because when we represent complex numbers as a plane, the multiplication of a complex number by i is equivalent to rotating the vector representing that number by an angle of $\scriptstyle \pi /2$ about the origin.

When Stephen Hawking wrote about "imaginary time" in his famous book A Brief History of Time, he was referring to Wick rotation.

Wick rotation also relates a QFT at a finite inverse temperature β to a statistical mechanical model over the "tube" R³×S¹ with the imaginary time coordinate τ being periodic with period β.

Note, however, that the Wick rotation cannot be viewed as a rotation on a complex vector space that is equipped with the conventional norm and metric induced by the inner product, as in this case the rotation would cancel out and have no effect at all.

Внешнии ссылки

• Wick rotation — a blog introduction
• A Spring in Imaginary Time — a worksheet in Lagrangian mechanics illustrating how replacing length by imaginary time turns the parabola of a hanging spring into the inverted parabola of a thrown particle