9

The definition of the newton is not a definition of force in general. It is the definition of a unit in the SI system. There is nothing circular or misleading there. Even using your quoted definition, the definition of the newton depends on the definition of force, but the definition of force does not depend on the definition of the newton. So that is linear,...


8

Given a four-vector $A^\mu$, define the “interval” associated with $A$ as $$ \Delta s_A = \eta_{\mu\nu}A^\mu A^\nu = \left(A^0\right)^2 -\vec A{}^2 $$ We say that $A$ is “spacelike” if $\Delta s_A < 0$. An example is $(0, \vec A)$. “timelike” if $\Delta s_A > 0$. An example is $(A^0, \vec 0)$. “lightlike” if $\Delta s_A = 0$. The “light cone” is the ...


8

Contact forces have no real distinction from electromagnetic forces, but that doesn't make the term irrelevant. It is shorter and easier to say "contact force" than to say "short range force based on un-modeled local electromagnetic interactions between nearby solids". So we use the term whenever it is convenient and whenever the meaning ...


6

The definition of the Newton unit is refering to the force magnitude only. As you say yourself, direction is not included in the mentioned definition - but remember that force can be considered both with and without direction. It depends on the scope, the frame, it is used within. Just as how a force can be considered two- or three-dimensional or more ...


5

The prefix names are for powers that are multiples of three as shown below. The $3.2$ is chosen randomly for the example. megacoulomb: $3.2\times10^6~\mathrm{C}=3.2~\mathrm{MC}$ kilocoulomb: $3.2\times10^3~\mathrm{C}=3.2~\mathrm{kC}$ Coulomb: $3.2\times10^0~\mathrm{C} = 3.2~\mathrm{C}$ millicoulomb: $3.2\times10^{-3}~\mathrm{C}=3.2~\mathrm{mC}$ microcoulomb: ...


5

Technically speaking, a propagating field in QFT is by definition a field with on-shell DOF, cf. e.g. this Phys.SE post. The terminology is inspired by wave propagation in the theory of waves.


5

I am not sure if this is what you are looking for, but nevertheless, let me say the following: In full generality, a "vielbein" on a d-dimensional spacetime is defined to be a vector bundle isomorphism $$e:T\mathcal{M}\to\mathcal{T},$$ where $T\mathcal{M}=\coprod_{p\in\mathcal{M}}T_{p}\mathcal{M}$ denotes the tangent bundle of the spacetime ...


4

Light cones can more generally be defined for a curved spacetime: Consider a $4$-dimensional Lorentzian manifold, that is, a $4$-dimensional smooth manifold $\mathcal{M}$ together with a metric $g$ with signature $(+,-,-,-)$. Then we can define for each point $p\in\mathcal{M}$ the "light-cone" $V_{p}\subset T_{p}\mathcal{M}$ to be the set of all &...


4

Gauge invariance is a red herring here that adds extra complication without helping address your question, so let me just focus on a scalar field $\phi(x)$ for simplicity. Loosely speaking, you can think of a field operator $\phi(x)$ as creating or annihilating a particle at position $x$. In this way, the time-ordered two point function $\langle 0 | T \phi(...


3

Upward force should be the weight of the liquid displaced by object, where did pressure came from? Mathematically the buoyant force is equal in magnitude to the weight of the displaced fluid, but that's not the main reason for buoyancy. If you had a container of water in deep space with no gravitational influence and you put an object in the water so that ...


2

The references for condensed matter are divided into solid-state physics and soft matter. Open any introductory book on any of those fields and you will see that neither gases nor plasma are considered (unless there is some interaction with a liquid/solid). The only exception to the rule is probably polymers, it may include the study of single molecules (but ...


2

Technically, when an object is immersed in a liquid it experiences an upward force known as Buoyant Force. This phenomenon of experiencing an upward force is known as Buoyancy. The origin of the buoyant force is actually pressure difference. Think of a cylindrical object fully submerged vertically in a liquid in a gravitational field. The liquid exerts ...


2

The velocity defined as $\vec{v}=\frac{d\vec{s}}{dt}$ is called instantaneous velocity. There is also average velocity which equals $\vec{v}=\frac{\Delta \vec{s}}{\Delta t}$, over some time $\Delta t$. In the case of uniform motion, average velocity over any time is the same as instantaneous velocity at any time. Uniform motion happens when there is no ...


2

Velocity is the rate of change in position (the derivative of position to time), $$\vec v=\frac{\mathrm d\vec s}{\mathrm dt}.$$ Acceleration is the rate of change in velocity (the double derivative of position to time), $$\vec a=\frac{\mathrm d\vec v}{\mathrm dt}=\frac{\mathrm d^2\vec s}{\mathrm dt^2}.$$ Basically, velocity is change in position while ...


2

With due respect to the formidable JunJohn S, his proof is doomed to unleash dyslexia demons of this kind, which I'd rather not deal with, and neither should you. From his (4.2.5), $$ \pi | x\rangle= |-x\rangle, $$ and $$ \hat p = \int\!\! dx ~|x\rangle (-i\hbar\partial_x)\langle x| , $$ you immediately see $$ \pi \hat p \pi^\dagger = \int\!\! dx |-x\...


2

What is time? A state that changes. There are a couple of problems here. You have to define what you allow to be a "state" and what you allow to vary when the state "changes" i.e. what is it changing with respect to. To illustrate these problems, suppose we define a state as "the average temperature at a given location on the ...


2

The formal definition of the flux of a vector field $\mathbf E$ through some surface $S$ is given by $$\iint_S\mathbf E\cdot\text d\mathbf a$$ where $\text d\mathbf a$ is a vector of magnitude equal to the area $\text da$ and direction normal to the surface $S$. So yes, you do need to consider the direction of the field. The "number of field lines" ...


2

You are right to be confused. Local gauge invariance has to do with locality, but not in an obviously direct way. Take the action for EM coupled to a charged scalar. $$\mathcal{L}_{gauge} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + (D_\mu\phi)^*D^\mu\phi,$$ where $F_{\mu\nu}(t,x) = \partial_\mu A_\nu(t,x) - \partial_\nu A_\mu(t,x)$ and $D_\mu\phi(t,x) = \partial_\...


2

The light cone is definied to be the set of 4-vectors $(ct,x,y,z)$ satisfying $$c^2t^2 - x^2 - y^2 - z^2 = 0.$$ Or written in covariant notation $$\eta_{\mu\nu} x^\mu x^\nu = 0.$$ (image from Einstein for Everyone - Spacetime)


2

As pointed out in the comment, The identity that you are trying to prove is mainly taken as the definition of a Hermitian operator. A is Hermitian if $A=A^\dagger$ or equivalently if $\langle x,Ay\rangle =\langle Ax,y\rangle \ \ \forall\ \ x,y.$ This can be seen as follows: $$\langle a|A^\dagger|b\rangle \equiv \langle b|A|a\rangle ^*$$ At this point, We can ...


2

We mame use of $$\partial_\mu \mathcal j^\mu=0 $$ $$\int d^3x (\partial_0\mathcal j^0 +\nabla\cdot \bf J)=0$$ for big enough volume the divergence term vanishes (by the divergence theorom $\int d^3x \nabla \cdot \bf J=\int \bf J \cdot \bf \hat n dA $ and the boundaries at infinity vanishes) giving you $$\frac{d}{dt}\int d^3x \mathcal j^0=0$$ Note that we ...


2

For the gravitational case, the definition of potential ($V$) is: $$V = \frac{U_g}{m}$$ Mathematically they represent different physical quantities. Potential energy is a property of a system of two masses, while potential is only a property of the source mass. For a spherical source mass $M$, the potential at a radial distance $r> a$ where $a$ is it's ...


1

Look at a solution of the (linearized) equation of motion for the pendulum, $$\theta(t)=A\cos\left(\sqrt{\frac{g}{l}}t+\varphi\right).$$ The amplitude $|A|$ is the maximum value of $\theta$; the angle varies between $-|A|$ and $|A|$ in the course of each period of length $T=2\pi\sqrt{l/g}$. The phase angle $\varphi$ tells you where in the cycle the pendulum ...


1

There are two related but different notions of "light cone" (or "null cone"): one in the spacetime [which Minkowski introduced in 1907/1908 as part of his "Space and Time", which introduced the "spacetime viewpoint" and the various terms we use today in relativity], and the other in the tangent space of a spacetime ...


1

According to the link, a linear directional effect is a physical thing that can be described by a polar vector: Force, velocity, electric field, and so on. Of course that answers circular with respect to the question, but that's it. It seems definitional, the only distinction is that one is a physical object and the other is a formal mathematical ...


1

You would be better to conflate your first two questions and define time as a continuum of non-spatial change, as the question 'what does time look like?' is literally meaningless-we cannot see time. Your statement about the arrow of time is false. High entropy does not rule out change over time. The idea of time having a direction is unnecessary- one can ...


1

You're right. A "Born" and "tree level" are the same thing. It's not very common to say Born anymore, but the reason why the call it like that in the reference is likely due to the more standard quantum mechanical definition. In nonrelativistic quantum mechanics one usually calls the "Born approximation" as the approximation at ...


1

Probably, the most general definition of a wave is a field evolving accordingly to an initial value problem for the field and its first n time derivatives ($n$ varies with the equation). Such a definition allows us to know if a particular partial differential equation corresponds to waves or not. Moreover, it is sufficiently general to include the solution ...


1

We can define a wave in the following way: There is a $y = f(x)$ over a domain. For example, the topography along a road. Now let's suppose another frame of reference travelling in the $-x$ direction, with a speed $-v$. It can be a car on the road. For the road's frame, the position of the car is $x_c = x_0 - vt$. In the frame of the car, that point is ...


1

This wave equation is linear. There are however non-linear extensions with non linear waves called kinks and solitons.


Only top voted, non community-wiki answers of a minimum length are eligible