(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$: $$ \int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i} $$

We can generalize this integral from a number $a$ to a matrix value $A$, from a number $x$ to a vector $X=(x_1,x_2,...,x_N)$.

So we generalize $$ \int_{-\infty}^{\infty} Dx \; e^{\frac{i}{2}X^T \cdot A \cdot X+ i JX } = \int_{-\infty}^{\infty} dx_1 \int_{-\infty}^{\infty} dx_2 \dots \int_{-\infty}^{\infty} dx_N e^{\frac{i}{2}X^T \cdot A \cdot X+ i JX }=\left(\frac{(2\pi i)^N}{\det(A)}\right)^{1/2} e^{-\frac{i}{2} J^T \cdot A^{-1} \cdot J}. \tag{ii} $$

(2) We can generalize again the integral to a functional integral or the so-called path integral in Quantum Field Theory.

So for a real scalar field $\phi$, can be regarded as the earlier Eq(i) real valued $x$ generalized to the field $$ x \to \varphi, $$ and the matrix value $A$ is generalized to the matrix operator $M$, so we can evaluate the path integral in the $N$-dimensional spacetime manifold $\Sigma_N$: $$ \int_{-\infty}^{\infty} D \varphi e^{i \int_{\Sigma_N} (d^N x)\big( -\frac{1}{2}\varphi \cdot M \cdot \varphi + J \cdot \varphi \big)} . \tag{iii} $$ where the Green's function $G(x_i-x_f)$ is a solution of the delta function $\delta^N(x_i-x_f)$ on right hand side (r.h.s): $$ -M G(x_i-x_f)=\delta^N(x_i-x_f). $$

Therefore, the $A$ matrix in Eq.(ii) is related to $M$, while also related to $G$ via the inverse generalization $$ A \to M, $$ $$ A^{-1}_{if}=G(x_i-x_f). $$ Thus we obtain:

$$ \int_{-\infty}^{\infty} D \varphi e^{i \int_{\Sigma_N} (d^N x)\big( -\frac{1}{2}\varphi \cdot M \cdot \varphi + J \cdot \varphi \big)}=Constant\cdot e^{\frac{-i}{2} \int d^N x_i \int d^N x_f J(x_i) G(x_i-x_f) J(x_f) }. \tag{iii} $$ involving the Green's function $G$. up to a not crucial constant $Constant$ which I do not care.

(3) Question:

Suppose I have a quartic integral from (i) to (i')

$$ \int_{-\infty}^{\infty} dx \; e^{h x - a x^2-b x^4}=?. \tag{i'} $$ I suppose that we can still easily get an answer.

How about the matrix value integral from (ii) to (ii'):

$$ \int_{-\infty}^{\infty} Dx \; e^{\frac{i}{2}X^T \cdot A \cdot X+ i J \cdot X +(X^T \cdot X)^2 } =?. \tag{ii'} $$ Do we also have a QFT path integral analogy: $$ \int_{-\infty}^{\infty} D \varphi e^{i \int_{\Sigma_N} (d^N x)\big( -\frac{1}{2}\varphi \cdot M \cdot \varphi + J \cdot \varphi + \varphi^4 \big)} =?. \tag{iii'} $$

My question concerns the neat evaluation of (i'), (ii') and (iii'). Thank you!

Here we can focus on the real valued function field $\varphi(x) \in \mathbb{R}$