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Tagged with path-integralregularization
56 questions
0votes
0answers
40views
Zeta-function regularization of constant product
I want to calculate a functional determinant coming from a Gaussian path integral with operator Matrix $M$. The determinant is given by the product over the eigenvalues according to $$\text{det}(M) = \...
1vote
1answer
108views
Is the Trotter formula justified in a theory that requires renormalization?
Usually, when QFT textbooks attempt to prove the equivalence of the path integral formulation with more familiar matrix mechanics, we make use of the Trotter formula. In Euclidean time, with $\hat{H}=\...
- 4,164
3votes
0answers
70views
Are measurable path integral Monte-Carlo correlation functions finite?
I am thinking about the correlation functions measured in path integral Monte-Carlo (PIMC) simulations. The Wick rotation $t \to -i\tau$ formulates the two-point correlation function $\langle \phi(...
- 700
2votes
1answer
89views
The path integral representation of the transition amplitude between the vacua of two field theories
The background of the question: PhysRevLett.115.261602. $ \newcommand\ket[1]{| #1 \rangle} \newcommand\braket[1]{\left\langle{#1}\right\rangle} \newcommand\dif{\mathrm{d}} \newcommand\E{\mathrm{e}} $...
- 120
1vote
2answers
71views
Infrared regularizing the harmonic oscillator path integral
This is from Laine and Vuorinen’s Basics of Thermal Field Theory. I do not understand why the fact that the integral over $x(\tau)$ implies the following regularization scheme. That is, I don’t ...
1vote
0answers
81views
Expression of $\langle 0 | 0 \rangle _{f,h}$ in the Srednicki's quantum field theory book (eq. (6.21), p.47) [duplicate]
I am reading the Srednicki's quantum field theory book and stuck at some statement. In the book p.46, the author worte that : "Now consider modifying the lagrangian of our theory by including ...
- 349
1vote
1answer
112views
Expectation value of the exponential of a quadratic term in fields
I have the following relation in this paper (J.B. Kogut: Introduction to Lattice Gauge Theory and Spin Systems, equation 8.39, page 709) (RG), where the author while doing an RG calculation writes $$\...
- 954
2votes
1answer
162views
How do Dedekind's eta function arise while computing the partition function of a compact scalar field over circle?
I am following the book String Theory in a nutshell (From Elias Kiritsis). In chapter 4.18, it takes a theory following the action: $$S=\frac{1}{4\pi l_s^2}\int X\square X\ d\sigma,\tag{4.18.1}$$ $$ \...
2votes
1answer
184views
How to integrate a Gaussian path integral of free particle using zeta function regularization?
I am attempting to integrate this path integral in Euclidean variable $\tau $ (but this need not be the same as the $X^0$ field): $$Z=\int _{X(0)=x}^{X(i)=x'}DX\exp \left(-\int _0^i d\tau \left[\frac{...
5votes
1answer
540views
The underlying cause of ill-defined loop-integrals in Quantum Field Theory
One of the main causes which leads to ill-defined loop integrals in Quantum Field Theory is that the variables of a Field Theory, $\varphi(x)$ for instance, are Quantum Fields which are governed by ...
- 10.4k
2votes
1answer
453views
Peskin and Schroeder's QFT eq. (9.14): Gaussian momentum field integration of phase space path integral
On Peskin and Schroeder's QFT book page 282, the book considered functional quantization of scalar field. First, begin with $$\left\langle\phi_b(\mathbf{x})\left|e^{-i H T}\right| \phi_a(\mathbf{x})\...
- 1,505
6votes
0answers
180views
Relationship between product integrals and functional determinants
This is in reference to the answer posted to this question. The person who answered the question claims that the functional determinant of any operator $O$ is given by a product integral $$\det O = \...
- 1,552
2votes
1answer
218views
Path integral with double integration involving the free particle case
Suppose we have the path integral: \begin{equation} Z=\int \mathcal{D}x\mathcal{D}y\,\exp\left[-\frac{a}{2}\int_0^1 dt\,\left(\,\dot{x}(t)^2-\,\dot{y}(t)^2\right)\right]. \end{equation} The ...
- 1,110
10votes
2answers
957views
Computing a Gaussian path integral with a zero-mode constraint
I have the following partition function: \begin{equation} Z=\int_{a(0)=a(1)} \mathcal{D}a\,\delta\left(\int_0^1 d\tau \,a -\bar{\mu}\right)\exp\left(-\frac{1}{g^2}\int_0^1d\tau\, a^2\right) \end{...
- 1,110
1vote
1answer
296views
Boundary conditions in Gaussian path integral
The $N$-dimensional Gaussian integral $$\int \mathrm{d}^N x \, \mathrm{e}^{-\frac{1}{2}\boldsymbol{x}^\mathrm{T}A\boldsymbol{x}+\boldsymbol{b}^\text{T}\boldsymbol{x}}=\left(\frac{(2\pi)^N}{\det A}\...
- 2,209
