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第1,209行： + + − 量子力学和概率之间的类比非常强烈，所以它们之间有许多数学上的联系。在一个离散时间的统计系统中，t = 1,2,3，由一个转移矩阵描述一个时间步长:+ 第1,218行：
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第1,273行： + + − 精细平衡表明，在平稳分布中从 m 到 n 的总概率，即从 m 开始的概率，乘以从 m 到 n 跳跃的概率，等于从 n 到 m 的概率，因此在平衡状态下的总的来回概率流是沿着任意跳跃的零。这个条件在 n = m 时自动得到满足，因此作为转移概率 r 矩阵的条件时，它具有同样的形式。+ − 第1,280行：
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第1,331行： − 在新的坐标系中，r 矩阵重新调整如下:+ 第1,336行：
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→Quantum mechanics in imaginary time虚时间量子力学
The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step <math>\scriptstyle K_{m\rightarrow n}</math>, the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:
The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step <math>\scriptstyle K_{m\rightarrow n}</math>, the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:
量子力学和概率论之间的类比性很强，因此它们之间有许多数学联系。在离散时间的统计系统中，t=1,2,3，由一个时间步的转移矩阵<math>\scriptstyle K_{m\rightarrow n}</math>描述，在有限个时间步之后，两点之间经过的概率可以表示为走每条路径的概率在所有路径上的和：
The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step <math>\scriptstyle K_{m\rightarrow n}</math>, the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:
The analogy between quantum mechanics and probability is very strong, so that there are many mathematical links between them. In a statistical system in discrete time, t=1,2,3, described by a transition matrix for one time step <math>\scriptstyle K_{m\rightarrow n}</math>, the probability to go between two points after a finite number of time steps can be represented as a sum over all paths of the probability of taking each path:
量子力学和概率论之间的类比性很强，因此它们之间有许多数学联系。在离散时间的统计系统中，t=1,2,3，由一个时间步的转移矩阵<math>\scriptstyle K{m\rightarrow n}</math>描述，在有限个时间步之后，两点之间经过的概率可以表示为走每条路径的概率在所有路径上的和：
:<math>
:<math>
<math>
<math>
K_{x\rightarrow y}(T) = \sum_{x(t)} \prod_t K_{x(t)x(t+1)}
K_{x\rightarrow y}(T) = \sum_{x(t)} \prod_t K_{x(t)x(t+1)}
\,</math>
\,</math>
where the sum extends over all paths <math>x(t)</math> with the property that <math>x(0)=0</math> and <math>x(T)=y</math>. The analogous expression in quantum mechanics is the [[Path integral formulation|path integral]].
where the sum extends over all paths <math>x(t)</math> with the property that <math>x(0)=0</math> and <math>x(T)=y</math>. The analogous expression in quantum mechanics is the [[Path integral formulation|path integral]].
其中，和扩展到所有路径<math>x(t)</math>，其属性为<math>x(0)=0</math> 和 <math>x(T)=y</math>。量子力学中的类似表达式是[[路径积分公式|路径积分]]。
where the sum extends over all paths <math>x(t)</math> with the property that <math>x(0)=0</math> and <math>x(T)=y</math>. The analogous expression in quantum mechanics is the path integral.
where the sum extends over all paths <math>x(t)</math> with the property that <math>x(0)=0</math> and <math>x(T)=y</math>. The analogous expression in quantum mechanics is the path integral.
A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys [[detailed balance]] when the stationary distribution <math>\rho_n</math> has the property:
A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys [[detailed balance]] when the stationary distribution <math>\rho_n</math> has the property:
概率论中的一般转移矩阵具有平稳分布，即无论起点是什么，在任何一点上找到的最终概率。如果任意两条路径同时到达同一点的概率为非零，则此平稳分布不依赖于初始条件。在概率论中，当平稳分布具有以下性质时，随机矩阵的概率m服从[[精细平衡]]：
A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys detailed balance when the stationary distribution <math>\rho_n</math> has the property:
A generic transition matrix in probability has a stationary distribution, which is the eventual probability to be found at any point no matter what the starting point. If there is a nonzero probability for any two paths to reach the same point at the same time, this stationary distribution does not depend on the initial conditions. In probability theory, the probability m for the stochastic matrix obeys detailed balance when the stationary distribution <math>\rho_n</math> has the property:
<math>
<math>
\rho_n K_{n\rightarrow m} = \rho_m K_{m\rightarrow n}
\rho_n K_{n\rightarrow m} = \rho_m K_{m\rightarrow n}
Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m <math>\rho_m</math> times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.
Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m <math>\rho_m</math> times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.
精细平衡表明，平稳分布中从m到n的总概率，即从m开始的概率乘以从m跳到n的概率，等于从n到m的概率，所以，在平衡状态下，沿着任何一个跳跃，总的来回概率为零。当n=m时，该条件自动满足，因此它在作为转移概率R矩阵的条件编写时具有相同的形式。
Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m <math>\rho_m</math> times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.
Detailed balance says that the total probability of going from m to n in the stationary distribution, which is the probability of starting at m <math>\rho_m</math> times the probability of hopping from m to n, is equal to the probability of going from n to m, so that the total back-and-forth flow of probability in equilibrium is zero along any hop. The condition is automatically satisfied when n=m, so it has the same form when written as a condition for the transition-probability R matrix.
精细平衡表明，平稳分布中从m到n的总概率，即从m开始的概率乘以从m跳到n的概率，等于从n到m的概率，所以，在平衡状态下，沿着任何一个跳跃，总的来回概率为零。当n=m时，该条件自动满足，因此它在作为转移概率R矩阵的条件编写时具有相同的形式。
<math>
<math>
\rho_n R_{n\rightarrow m} = \rho_m R_{m\rightarrow n}
\rho_n R_{n\rightarrow m} = \rho_m R_{m\rightarrow n}
\,</math>
\,</math>
When the R matrix obeys detailed balance, the scale of the probabilities can be redefined using the stationary distribution so that they no longer sum to 1:
When the R matrix obeys detailed balance, the scale of the probabilities can be redefined using the stationary distribution so that they no longer sum to 1:
当 r 矩阵满足精细平衡时，可以使用平稳分布重新定义概率的大小，使它们不再和为1:
当 R 矩阵满足精细平衡时，可以使用平稳分布重新定义概率的大小，使它们不再和为1:
<math>
<math>
p'_n = \sqrt{\rho_n}\;p_n
p'_n = \sqrt{\rho_n}\;p_n
\,</math>
\,</math>
In the new coordinates, the R matrix is rescaled as follows:
In the new coordinates, the R matrix is rescaled as follows:
在新的坐标系中，R 矩阵重新调整如下:
<math>
<math>
\sqrt{\rho_n} R_{n\rightarrow m} {1\over \sqrt{\rho_m}} = H_{nm}
\sqrt{\rho_n} R_{n\rightarrow m} {1\over \sqrt{\rho_m}} = H_{nm}
\,</math>
\,</math>
<math>
<math>
H_{nm} = H_{mn}
H_{nm} = H_{mn}
\,</math>
\,</math>
This matrix H defines a quantum mechanical system:
This matrix H defines a quantum mechanical system:
这个矩阵 h 定义了一个量子力学系统:
这个矩阵 H 定义了一个量子力学系统:
<math>
<math>
i{d \over dt} \psi_n = \sum H_{nm} \psi_m
i{d \over dt} \psi_n = \sum H_{nm} \psi_m
\,</math>
\,</math>
whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The [[eigenvectors]] are the same too, except expressed in the rescaled basis. The stationary distribution of the statistical system is the ''ground state'' of the Hamiltonian and it has energy exactly zero, while all the other energies are positive. If H is exponentiated to find the U matrix:
其哈密顿量与统计系统的R矩阵的特征值相同。[[特征向量]]也是相同的，除了在重新缩放的基础上表示。统计系统的平稳分布是哈密顿量的“基态”，它的能量正好为零，而所有其他能量都为正。如果H被指数化以找到U矩阵：
whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The eigenvectors are the same too, except expressed in the rescaled basis. The stationary distribution of the statistical system is the ground state of the Hamiltonian and it has energy exactly zero, while all the other energies are positive. If H is exponentiated to find the U matrix:
whose Hamiltonian has the same eigenvalues as those of the R matrix of the statistical system. The eigenvectors are the same too, except expressed in the rescaled basis. The stationary distribution of the statistical system is the ground state of the Hamiltonian and it has energy exactly zero, while all the other energies are positive. If H is exponentiated to find the U matrix:
其哈密顿量与统计系统的 r 矩阵具有相同的特征值。特征向量也是相同的，除了用重标度基表示。统计系统的稳态分布是哈密顿量的基态，它的能量精确为零，而其它所有能量都是正的。如果 h 是求 u 矩阵的幂:
其哈密顿量与统计系统的 R 矩阵具有相同的特征值。特征向量也是相同的，除了用重标度基表示。统计系统的稳态分布是哈密顿量的基态，它的能量精确为零，而其它所有能量都是正的。如果 H 是求 U 矩阵的幂:
<math>
<math>
U(t) = e^{-iHt}
U(t) = e^{-iHt}
\,</math>
\,</math>
<math>
<math>
K'(t) = e^{-Ht}
K'(t) = e^{-Ht}
\,</math>
\,</math>
For quantum systems which are invariant under [[T-symmetry|time reversal]] the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on [[supersymmetry]].
For quantum systems which are invariant under [[T-symmetry|time reversal]] the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on [[supersymmetry]].
对于在[[T-对称|时间反转]]下不变的量子系统，哈密顿量可以是实的和对称的，因此时间反转对波函数的作用仅仅是复共轭。如果这样的哈密顿量有一个唯一的具有正实波函数的最低能量态，就像它经常由于物理原因所做的那样，它在虚时间内与一个随机系统相连。随机系统和量子系统之间的这种关系为[[超对称性]]提供了很多线索。
For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on supersymmetry.
For quantum systems which are invariant under time reversal the Hamiltonian can be made real and symmetric, so that the action of time-reversal on the wave-function is just complex conjugation. If such a Hamiltonian has a unique lowest energy state with a positive real wave-function, as it often does for physical reasons, it is connected to a stochastic system in imaginary time. This relationship between stochastic systems and quantum systems sheds much light on supersymmetry.