Gaussian integral for a vector and a function - how to evaluate

Question

My problem concerns evaluation of a Gaussian integral. Let there be a real vector $\mathbf{v}$ and a matrix $\mathbf{A}$. I would like to know the result of the following integral: $$ \int_{-\infty}^{\infty} \mathrm{d}v_1 \mathrm{d}v_2...\mathrm{d}v_N \, \exp \left ( - \frac{1}{2} \mathbf{v^T} \mathbf{A} \mathbf{v} \right ) = \int_{-\infty}^{\infty} \mathrm{d}v_1 \mathrm{d}v_2...\mathrm{d}v_N \, \exp \left ( - \frac{1}{2} \sum_{i=1}^{N} \sum_{j=1}^{N} v_i A_{ij} v_j \right ) \, . $$ I encountered such integrals in the context of Feynman path integration in quantum field theory. (For sure there are requirements on $\mathbf{A}$, however, I am more concerned with the principle.)

My approach follows. I look at eigenvectors $\mathbf{u}$ and eigenvalues $\lambda$ of the matrix defined as $$ A_{ij} u_j^n = \lambda^n u_i^n \, , $$ for the $n$th eigenvector/value. I then decompose the vector $\mathbf{v}$ into a linear combination of the eigenvectors as $$ v_i = \sum_{n = 1}^{N} \alpha^n u_i^n \, . $$ Then $$ A_{ij} v_j = A_{ij} \sum_{n = 1}^{N} \alpha^n u_j^n = \sum_{n = 1}^{N} \alpha^n \lambda^n u_i^n $$ Therefore the term $\mathbf{v^T} \mathbf{A} \mathbf{v}$ turns into $\sum_{n = 1}^{N} (\alpha^n)^2 \lambda^n$. That is promising, since the Gaussian integral now decouples into only diagonal terms.

The problematic bit for me is the evaluation of $\mathrm{d}v_1 \mathrm{d}v_2...\mathrm{d}v_N$ in the new coordinates $\{\alpha^n\}$. I could simply write $$ \mathrm{d}\alpha^1 \mathrm{d}\alpha^2...\mathrm{d}\alpha^N = \det \left ( \frac{\partial(\alpha^1,\alpha^2,...,\alpha^N)}{\partial(v_1,v_2,...,v_N)} \right ) \mathrm{d}v_1\mathrm{d}v_2...\mathrm{d}v_N $$ using the Jacobian. But now I got stuck. According to my textbook, the Gaussian integral should evaluate to $$ \int_{-\infty}^{\infty} \mathrm{d}v_1 \mathrm{d}v_2...\mathrm{d}v_N \, \exp \left ( - \frac{1}{2} \mathbf{v^T} \mathbf{A} \mathbf{v} \right ) = \frac{(2 \pi)^{N/2}}{\det(\mathbf{A})^{1/2}}\, . $$ Could you please help me to get there?

Functional form: I would then like to look at the limit of $N \to \infty$ and promote the vector $\mathbf{v}$ to a function and the matrix $\mathbf{A}$ to an operator. I am interested in the integral $$ \int_{-\infty}^{\infty} \mathrm{D}f \, \exp \left ( \int_a^b \mathrm{d} \, t f(t) \hat{A} f(t) \right ) $$ and how to make contact between it and the vectorial form.

Thanks a lot.

SSF

Answer 1

I am going to assume $\mathbf{A}$ is symmetric. Then it can be diagonalised by an orthogonal matrix $\mathbf{P}$ so that $\mathbf{A}=\mathbf{P^T B P}$, where $\mathbf{B}$ is diagonal. Substitute it into the integral and then make the substitution $\mathbf{u}=\mathbf{Pv}$. The Jacobian is $\mathbf{\det{P}}=1$. $$ \int_{-\infty}^{\infty} \mathrm{d}v_1 \mathrm{d}v_2...\mathrm{d}v_N \, \exp \left ( - \frac{1}{2} \mathbf{v^T} \mathbf{A} \mathbf{v} \right ) = \int_{-\infty}^{\infty} \mathrm{d}u_1 \mathrm{d}u_2...\mathrm{d}u_N \, \exp \left ( - \frac{1}{2} \sum_{i=1}^{N} \lambda_i u_i^2 \right ) \, , $$ where the $\lambda_i$ are the eigenvalues of $\mathbf{A}$ and thus the diagonal elements of $\mathbf{B}$. Then you just separate the independent integrals into a product of regular Gaussian integrals with the standard result. Note that $$ \prod_i \lambda_i=\mathbf{\det{A}}\,. $$