# What is Scalar fields: Definition and 40 Discussions

In mathematics and physics, a scalar field or scalar-valued function associates a scalar value to every point in a space – possibly physical space. The scalar may either be a (dimensionless) mathematical number or a physical quantity. In a physical context, scalar fields are required to be independent of the choice of reference frame, meaning that any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory.

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16. ### A How to derive general solution to the Klein-Gordon equation

I understand that the ansatz to $$(\Box +m^{2})\phi(\mathbf{x},t)=0$$ (where ##\Box\equiv\partial^{\mu}\partial_{\mu}=\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}##) is of the form ##\phi(\mathbf{x},t)=e^{(iE_{\mathbf{k}}t-\mathbf{k}\cdot\mathbf{x})}##, where...
17. ### A Evolution of Scalar Field: Equation Demonstration

I'm looking for a demonstration of the equation governing the evolution of the scalar field: ## \Box \phi = \frac{1}{\sqrt{g}} \frac{ \partial}{\partial x^{\mu}} \sqrt(g)g^{(\mu)(\nu)} \frac{\partial}{\partial x^{\nu}} \phi=0## I used the lagrangian for a scalar field: ## L = \nabla_{\mu}\phi...
18. ### A Canonical quantization of scalar fields

In the srednicki notes he goes from $$H = \int d^{3}x a^{\dagger}(x)\left( \frac{- \nabla^{2}}{2m}\right) a(x)$$ to $$H = \int d^{3}p\frac{1}{2m}P^{2}\tilde{a}^{\dagger}(p)\tilde{a}(p)$$ Where $$\tilde{a}(p) = \int \frac{d^{3}x}{(2\pi)^{\frac{3}{2}}}e^{-ipx}a(x)$$ Is this as simple as...
19. ### SR & Lorentz Scalar Fields: Covariant Diff. & Wave Amplitude

Hi. In GR , covariant differentiation is used because the basis vectors are not constant. But , what about in SR ? If the basis vectors are not Cartesian then they are not constant. Does covariant differentiation exist in SR ? And are for example spherical polar basis vectors which are not...
20. ### Scalar fields and the Higgs boson

This is more of a QFT question, so the moderator may want to move it to another forum. The simplest example of a QFT that I learned was the scalar field; in Sakurai's 1967 textbook. I know the Higgs is a J=0 particle. Is it described by the simple scalar field discussed in Sakurai's text? I ask...
21. ### Time-ordered product of real scalar fields

Hi guys, So I've got a real scalar field which is the sum of the positive frequency part and negative frequency part: \phi(x)=\phi^{(+)}(x)+\phi^{(-)}(y) and I'm looking at the time-ordered product: T(\phi(x)\phi(y))=\theta(x^{0}-y^{0})\phi(x)\phi(y)+\theta(y^{0}-x^{0})\phi(y)\phi(x) for...
22. ### Lorentz transforming differential operators on scalar fields

Homework Statement I'm reading Peskin and Schroeder to the best of my ability. Other than a few integration tricks that escaped me I made it through chapter 2 with no trouble, but the beginning of chapter three, "Lorentz Invariance in Wave Equations", has me stumped. They are going through a...
23. ### Direction of the maximum gradient (scalar fields)

If a question asks for the direction of the maximum gradient of a scalar field, is it acceptable to just use del(x) as the answer or is the question asking for a unit vector? Thanks
24. ### Understanding Scalar Fields: Tools for Studying Vector Field Behavior

After watch this video , I understood that for study the behavior of the vector field, just use 2 tools, the line integral and the surface integral, and actually too, the divergence and the curl. In accordance with this, the maxwell's equations are justly the line integral, the surface integral...
25. ### Non-canonical terms of scalar fields

Hello! Well, I guess it's all in the title, really. I was reading about k-essence, and it was described as a scalar field having a non-canonical kinetic term. I did a bit of browsing and couldn't find a clear explanation of what, exactly, a non-canonical kinetic term is. Any help would be...
26. ### Question on derivatives of Hermitian conjugate scalar fields

Hi, I know this question may seem a little trivial, but is there any real difference between \left (\partial_{\mu} \phi \right)^{\dagger} and \partial_{\mu} \phi^{\dagger} and if so, could someone provide an explanation? Many thanks. (Sorry if this isn't quite in the right...
27. ### Why don't scalar fields propagate superluminally?

This is a really basic question, but... Say I have a massive scalar field obeying the Klein-Gordon equation linearized about flat space, \partial_t^2 \phi + (k^2 + m^2)\phi = 0. This has solutions \phi \sim e^{\pm \sqrt{k^2 + m^2}t} and the sound speed should be \omega_k/k =...
28. ### Scalar fields/ Scalar functions / Vector fields / Vector functions

I know that physically, they describe relationships whereby, for instance a vector field, for each point in three dimensional space (a "vector"), we have a "vector" which has a direction or magnitude. Now I once asked what the difference between a vector field and a vector function is and the...
29. ### Integral in Commutator of Scalar fields

So, in the calculation of D(t,r) = \left[ \phi(x) , \phi(y) \right] , where t= x^0 - y^0,~ \vec{r} = \vec{x} - \vec{y} you need to calculate the following integral D(t,r) = \frac{1}{2\pi^2 r} \int\limits_0^\infty dp \frac{ p \sin(p r) \sin \left[(p^2 + m^2)^{1/2} t \right]} { (p^2 + m^2...
30. ### Understanding the Fine Tuning Problem in Scalar Field Theories

What do people mean when they say that mass renormalization of scalar field theories confronts us with a fine tuning problem. It's said the divergence in the mass of a scalar field is quadartic, rather than logarithmic, this poses a fine tuning problem. Why and how, and what does that mean...

Curl div... Homework Statement f is a scalar field. What does div(f) curl(f) rotgrad(f) divgrad(f) stand for? I need to know if a scalar field can have the meanings of roration and diverge like a vector field
32. ### Scalar fields: why symmetric ener-mom. tensor?

I'm studying the properties of the energy momentum tensor for a scalar field (linked to the electromagnetic field and corresponding energy-momentum tensor) and now I'm facing the statement: "for a theory involving only scalar fields, the energy-momentum tensor is always symmetric". But I've...
33. ### Functional Quantization of Scalar Fields

Hi everyone, I'm reading section 9.2 of Peskin and Schroeder, and have trouble understanding the origin of a term in the transition from equation 9.26 to 9.27. Specifically, equation 9.26 is \frac{1}{V^2}\sum_{m,l}e^{-(k_m\cdot x_1 + k_l\cdot x_2)}\left(\prod_{k_{n}^{0}>0}\int d \Re...
34. ### Lorentz Transformation of Scalar Fields

Homework Statement Working on an exercise from Srednicki's QFT and something is not clear. Show that [\varphi(x), M^{uv}] = \mathcal{L}^{uv} \varphi(x) where \mathcal{L}^{uv} = \frac{\hbar}{i} (x^u \partial^v - x^v \partial^u ) Homework Equations (1) U(\Lambda)^{-1} \varphi(x)...
35. ### Understanding Scalar Fields: Assigning Values to Space

"A scalar field assigns every point in space to a scalar value" Would this be a correct definition of a scalar field? Thanks
36. ### Interactions of Fermion & Scalar Fields: Exploring the Difference

Suppose I couple a fermion field to a scalar field using \mathrm{i} g \bar{\psi}\psi \varphi and \mathrm{i} g \bar{\psi}\gamma_5\psi\varphi. I'm trying to understand what would be the physical difference between these interactions. I know that (1/2)(1\pm \gamma_5) approximately projects out...
37. ### DEC with E/M and scalar fields

Hi, I'm trying to show that electromagnetism and scalar field theories satisfy the DEC. I know how to find T_{\mu\nu} and all that and what I have to show (T_{\mu\nu} T^\nu_{\ \lambda} t^\mu t^\lambda\leq 0 and T_{\mu\nu} t^\mu t^\nu\geq 0 for timelike t^\mu), but I'm having trouble getting...
38. ### How Do Baryons Emerge in the Standard Model Through Topological Effects?

I'm lazy so I'm going to start bringing my questions here. Correct me if I'm wrong, but isn't it true that baryons only enter the standard model through this subtle topological effect. Now this is where I'm at. I kind of got the Goldstone boson concept, maybe someone could better explain...
39. ### without resorting to scalar fields

"...without resorting to scalar fields" http://arxiv.org/astro-ph/0703566 Co-authored by Parampreet Singh, one of the experts in Quantum Cosmology (gauged by publication trackrecord and citations by other scholars, see: https://www.physicsforums.com/showthread.php?p=1368143#post1368143 )...
40. ### Understanding Composite Fields and Scalar Fields

Dear PF, Can I consider the composite field for instance psi_bar psi as a scalar filed? I mean can it be the same in all respects? Can this composite field and scalar filed treated as totally equivalent? Thks