Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
Next revision Both sides next revision
magic_gaussian_integral [2015/04/15 13:43]
nikolaj
magic_gaussian_integral [2016/12/06 16:43]
nikolaj
Line 1: Line 1:
 +===== Magic Gaussian integral =====
 +==== Partial function ====
 +
 +==== Discussion ====
 +The $\varepsilon$ prescription in the definition is just there so that one can evaluate the integral for certain complex matrices $A$ where it wouldn'​t exist otherwise. For example if $A$ has imaginary eigenvalues,​ then the naive integral will not be finite, while if we use $A_\varepsilon:​=A-\varepsilon\,​\mathrm{1}$,​ then we get an additional term $\mathrm{e}^{-\varepsilon\,​\left\langle\phi\left|\right.\phi\right\rangle}$ which makes the integral converge.
 +
 +That whole integral is just a showoff generalization of the one dimensional integral ​
 +
 +^ $I_a:​=\int_{-\infty}^\infty{\mathrm e}^{-\tfrac{1}{2}a\,​\phi^2}\mathrm d\phi =(2\pi)^{1/​2}a^{-1/​2}$ ^
 +
 +which is obtained, for example, with the basic trick of switching to polar coordinates. ​
 +
 +(Side remark: Having found the normalization of ${\mathrm e}^{-\tfrac{1}{2}a\,​\phi^2}$ lets us define the function ${\mathrm{erf}}$ with $\lim_{d\to\infty}{\mathrm{erf}}(d)=1$. Then we can show that $\frac{1}{2}\pi^\frac{1}{2}{\mathrm{erf}}(d) = \sum_{k=0}^\infty\frac{1}{k!}(-1)^k\frac{d^{2k+1}}{2k+1}$ and with $\int_0^d(-\phi^2)^k{\mathrm d}\phi = (-1)^k\frac{d^{2k+1}}{2k+1}$ we can evaluate the initial integral on $[-d,d]$, not just all of $\mathbb R$.)
 +
 +Adding a imaginary linear term in the exponent results in the statement that the Fourier transform of a Gaussian curve is again a Gaussian curve
 +
 +^ $\int_{-\infty}^\infty\mathrm e^{-\tfrac{1}{2}a\,​\phi^2+i\,​\phi\,​j}\mathrm d\phi =I_a\cdot\mathrm e^{-\tfrac{1}{2}j^2\,​a^{-1}}$ ^
 +
 +which can be obtained via an integration variable shift.
 +
 +=== Theorems ===
 +
 +| @#55CCEE: $ A_\varepsilon $ ... $n\times n$ square matrix over $\mathbb C$ | @#55CCEE: $ \lim_{\varepsilon\to 0}A_\varepsilon = A $ |
 +| @#FFBB00: $Z_f(J):​=\lim_{\varepsilon\to 0}\int_{-\infty}^\infty\,​f(\phi)\,​\mathrm e^{\frac{1}{2} \left\langle\phi\left|\,​A_\varepsilon\,​\right|\phi\right\rangle +i\,​\left\langle\phi\left|\right.J\right\rangle}\prod_{i=1}^m \mathrm d\phi_i $ | @#FFBB00: |
 +
 +Here $\langle u|v\rangle$ denotes the inner product in ${\mathbb C}^n$ as vector space.
 +
 +Notice that via diagonalization of the matrix and knowledge of basic Gaussian integral above, we get
 +
 +^ $Z_1(0):​=\int_{-\infty}^\infty \mathrm e^{-\tfrac{1}{2}\left\langle\phi\left|\,​A\,​\right|\phi\right\rangle} \prod_{i=1}^m \mathrm d\phi_i = (2\pi)^{1/​2}(\det A)^{-1/2} $ ^
 +
 +Taking care of the vector $J$, we can obtain ​
 +
 +^ $Z_1(J):​=\int_{-\infty}^\infty \mathrm e^{-\tfrac{1}{2}\left\langle\phi\left|\,​A\,​\right|\phi\right\rangle +i\,​\left\langle\phi\left|\right.J\right\rangle}\prod_{i=1}^m\mathrm d\phi_i = Z_1(0)\cdot\mathrm e^{-\frac{1}{2}\left\langle J\left|\,​A^{-1}\,​\right|J\right\rangle }$ ^
 +
 +Now from a physical perspective,​ it's actually better to write this as $\propto\mathrm e^{-\frac{1}{2}\left\langle A^{-1}J\left|\,​A\,​\right|A^{-1}J\right\rangle }$. In the path integral treatment of the diffusion equation, the propagator involves quantity $\phi\equiv p,J\equiv q$ and roughly speaking $A\propto \delta t,\ A^{-1}\propto\tfrac{\mathrm d}{\mathrm dt}$. The above integral plays a role in passing from the Hamiltonian perspective to the Lagrangian one: $\left\langle\phi\left|\,​A\,​\right|\phi\right\rangle\propto p^2\Delta t$ to a sort of conjugate $\left\langle A^{-1}J\left|\,​A\,​\right|A^{-1}J\right\rangle\propto {\dot q}^2\Delta t$.
 +
 +Lastly, notice that $-i\frac{\partial}{\partial J_i}e^{i\,​\left\langle\phi\left|\right.J\right\rangle}=\phi_i\,​ \mathrm e^{i\,​\left\langle\phi\left|\right.J\right\rangle}$ and therefore
 +
 +^ $ Z_f(J) = Z_1(0)\ f\left(-i\frac{\partial}{\partial J}\right)\,​\mathrm e^{-\frac{1}{2}\left\langle J\left|\,​A^{-1/​2}\,​\right|J\right\rangle }$ ^
 +
 +
 +We are interested in that expression as the solution of the integral, because in quantum field theory, the path integral is often an infinite dimensional variant it. There the exponent in the defining integral is the action functional, the operator $A$ involves a hard to invert differential operator (the inverse being strongly related to the response function/​green function) and $f$ encodes the type of process and the interaction. The terms from the expansion of $f$ are encoded by Feynman diagrams. ​
 +
 +=== Reference ===
 +Wikipedia: [[http://​en.wikipedia.org/​wiki/​Common_integrals_in_quantum_field_theory|Common integrals in quantum field theory]], [[http://​en.wikipedia.org/​wiki/​Gaussian_integral|Gaussian integral]]
 +==== Parents ====
 +=== Context ===
 +[[Square matrix]], [[Self-adjoint operator]]
 +=== Requirements ===
 +[[Function integral on ℝⁿ]], [[Exponential function]]
 +=== Related ===
 +[[Retarded propagator . time-independent Hamiltonian]]
 +
 ===== Magic Gaussian integral ===== ===== Magic Gaussian integral =====
 ==== Partial function ==== ==== Partial function ====
Line 38: Line 92:
 Lastly, notice that $-i\frac{\partial}{\partial J_i}e^{i\,​\left\langle\phi\left|\right.J\right\rangle}=\phi_i\,​ \mathrm e^{i\,​\left\langle\phi\left|\right.J\right\rangle}$ and therefore Lastly, notice that $-i\frac{\partial}{\partial J_i}e^{i\,​\left\langle\phi\left|\right.J\right\rangle}=\phi_i\,​ \mathrm e^{i\,​\left\langle\phi\left|\right.J\right\rangle}$ and therefore
  
-^ $ Z_f(J) = f\left(-i\frac{\partial}{\partial J}\right)\, ​Z_1(J$ ^+^ $ Z_f(J) = f\left(-i\frac{\partial}{\partial J}\right)\,\mathrm e^{-\frac{1}{2}\left\langle ​J\left|\,​A^{-1/​2}\,​\right|J\right\rangle } $ ^
  
 i.e. i.e.
Line 47: Line 101:
  
 === Reference === === Reference ===
-Wikipedia: [[http://​en.wikipedia.org/​wiki/​Common_integrals_in_quantum_field_theory|Common integrals in quantum field theory]], [[http://​en.wikipedia.org/​wiki/​Gaussian_integral|Gaussian integral]]+Wikipedia: ​ 
 +[[http://​en.wikipedia.org/​wiki/​Common_integrals_in_quantum_field_theory | Common integrals in quantum field theory]], ​ 
 +[[http://​en.wikipedia.org/​wiki/​Gaussian_integral | Gaussian integral]]
 ==== Parents ==== ==== Parents ====
 === Context === === Context ===
Link to graph
Log In
Improvements of the human condition