It's about how to evaluate the integral below, $$ D(x-y)=\int \frac{d^3 p}{(2\pi)^3}\frac{1}{2\sqrt{p^2+m^2}}e^{-i\vec p\cdot (\vec x-\vec y)} $$ it describes the amplitude for a scalar particle in free Klein-Gordon field propagating from y to x (y and x have a space-like interval) . The book said that the result can be written explicitly in terms of Bessel function but I have not idea at all.
2026-03-26 06:04:10.1774505050
Problems 2.3 in Peskin's book
532 Views Asked by Fan https://math.techqa.club/user/fan/detail AtRelated Questions in INTEGRATION
- How can I prove that $\int_0^{\frac{\pi}{2}}\frac{\ln(1+\cos(\alpha)\cos(x))}{\cos(x)}dx=\frac{1}{2}\left(\frac{\pi^2}{4}-\alpha^2\right)$?
- How to integrate $\int_{0}^{t}{\frac{\cos u}{\cosh^2 u}du}$?
- Show that $x\longmapsto \int_{\mathbb R^n}\frac{f(y)}{|x-y|^{n-\alpha }}dy$ is integrable.
- How to find the unit tangent vector of a curve in R^3
- multiplying the integrands in an inequality of integrals with same limits
- Closed form of integration
- Proving smoothness for a sequence of functions.
- Random variables in integrals, how to analyze?
- derive the expectation of exponential function $e^{-\left\Vert \mathbf{x} - V\mathbf{x}+\mathbf{a}\right\Vert^2}$ or its upper bound
- Which type of Riemann Sum is the most accurate?
Related Questions in CONTOUR-INTEGRATION
- contour integral involving the Gamma function
- Find contour integral around the circle $\oint\frac{2z-1}{z(z-1)}dz$
- prove that $\int_{-\infty}^{\infty} \frac{x^4}{1+x^8} dx= \frac{\pi}{\sqrt 2} \sin \frac{\pi}{8}$
- Intuition for $\int_Cz^ndz$ for $n=-1, n\neq -1$
- Complex integral involving Cauchy integral formula
- Contour integration with absolute value
- Contour Integration with $\sec{(\sqrt{1-x^2})}$
- Evaluating the integral $\int_0^{2\pi}e^{-\sqrt{a-b\cos t}}\mathrm dt$
- Integral of a Gaussian multiplied with a Confluent Hypergeometric Function?
- Can one solve $ \int_{0}^\infty\frac{\sin(xb)}{x^2+a^2}dx $ using contour integration?
Related Questions in QUANTUM-FIELD-THEORY
- Conserved currents under Lorentz Transformations
- Yang–Mills theory and mass gap
- Complex Gaussian integral
- Quantum invariants of 2-knots
- QFT, more a QM Question, Hamiltonian relation time evolution
- Spinor chiral transformation by $\psi \to \gamma^5 \psi$
- How are topological invariants obtained from TQFTs used in practice?
- Computing energy-momentum tensor with arrowed Feynman slash derivative
- Feynman Vertex Rule, Correlator, 2 diff coupling constants, why no-cross terms
- QFT, Noether and Invariance, Complex fields, Equal mass
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?