Newest questions tagged definition - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-07-19T02:06:31Z https://physics.stackexchange.com/feeds/tag?tagnames=definition&sort=newest http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://physics.stackexchange.com/q/492271 0 Specific total enthalpy VS Specific enthalpy user29463 https://physics.stackexchange.com/users/234851 2019-07-18T07:35:35Z 2019-07-18T09:26:27Z <p>What is the difference between specific enthalpy and specific total enthalpy in the context of fluid flow?</p> https://physics.stackexchange.com/q/491915 1 What is a mass moment? Aidanaidan12 https://physics.stackexchange.com/users/234464 2019-07-16T13:04:10Z 2019-07-16T16:19:52Z <p>I am currently reading through a document <a href="https://ocw.mit.edu/courses/mechanical-engineering/2-003j-dynamics-and-control-i-spring-2007/lecture-notes/lec11.pdf" rel="nofollow noreferrer">Finding Moments of Inertia</a> from MIT, page 4, and I am a little confused as to one of the concepts that they use.</p> <p>In this document, there is mention of a <a href="https://www.google.com/search?as_epq=mass+moment" rel="nofollow noreferrer">mass moment</a>. Could someone possibly define this for me please? I can't find anything too clear on the Internet.</p> <p>Is this synonymous with the first moment of mass?</p> https://physics.stackexchange.com/q/491788 1 A concise definition of a frame of reference in Newtonian mechanics? booNlatoT https://physics.stackexchange.com/users/236935 2019-07-15T17:31:41Z 2019-07-15T21:04:58Z <p>I've read <a href="https://en.wikipedia.org/wiki/Frame_of_reference" rel="nofollow noreferrer">Wikipedia's entry</a> on frame of reference and also followed all of the references cited in the text (Salençon, Brillouin, Norton, etc) but I'm struggling to find any concise definition in all of that. </p> <p>I would like a concise definition for a frame of reference in the context of Newtonian mechanics. This definition should not involve any additional qualifiers such as inertial and must be mature enough as to differentiate a frame of reference and a coordinate system. Is there one such definition?</p> https://physics.stackexchange.com/q/491765 0 Definition of non-conservative force [duplicate] curious https://physics.stackexchange.com/users/195374 2019-07-15T15:40:54Z 2019-07-15T15:53:29Z <div class="question-status question-originals-of-duplicate"> <p>This question already has an answer here:</p> <ul> <li> <a href="/questions/31672/what-causes-a-force-field-to-be-non-conservative" dir="ltr">What causes a force field to be &ldquo;non-conservative?&rdquo;</a> <span class="question-originals-answer-count"> 2 answers </span> </li> </ul> </div> <p>In defining <a href="https://en.wikipedia.org/wiki/Conservative_force" rel="nofollow noreferrer">conservative force</a>, we say that</p> <blockquote> <p>"The potential energy difference is path independent."</p> </blockquote> <p>However, as far as I understand, potential energy only exists when there is a force field.</p> <p>People say one example of non-conservative force.</p> <p>By definition, non-conservative force should be the one in which the difference in potential energy is path dependent. But where is potential energy for friction which is not a force field?</p> https://physics.stackexchange.com/q/491530 7 Equivalent definitions of total angular momentum TheoreticalMinimum https://physics.stackexchange.com/users/224659 2019-07-14T05:32:14Z 2019-07-14T17:15:39Z <p>Consider the equality <span class="math-container">\begin{equation}\exp\left(-\frac{i}{\hbar}\boldsymbol{\phi J}\right)\left|x\right&gt;=\left|R(\phi)x\right&gt;,\end{equation}</span> where <span class="math-container">$\left|x\right&gt;$</span> denotes a position eigenstate, <span class="math-container">$J$</span> the total angular momentum operator on the ket space, and <span class="math-container">$R(\boldsymbol{\phi})$</span> the <span class="math-container">$\mathbb{R}^3$</span> rotation matrix for rotations around <span class="math-container">$\boldsymbol{\phi}=\phi\cdot\mathbf{n}$</span> with <span class="math-container">$||\mathbf{n}||=1$</span>.</p> <p>Some consider this equation to be the definition of total angular momentum. How can this equation be proven by using the arguably more popular and classically motivated definition? <span class="math-container">\begin{equation} \mathbf{J}:=\mathbf{L}+\mathbf{S}=\mathbf{X}\times\mathbf{P}+\mathbf{S} \end{equation}</span></p> <p><strong>Note:</strong> Thanks to the work of @Valter Moretti and @Adam Latosiński an equality between the two defintions has been established for a spinless particle. </p> https://physics.stackexchange.com/q/491288 1 Definition of "specific gravity" Thomas https://physics.stackexchange.com/users/141461 2019-07-12T21:37:10Z 2019-07-13T05:37:43Z <p>I've learnt that a <strong>specific quantity</strong> is an extensive quantity divided by the mass. How does the definition of <a href="https://en.wikipedia.org/wiki/Specific_gravity" rel="nofollow noreferrer"><strong>specific gravity</strong></a> fit into this scheme?</p> https://physics.stackexchange.com/q/491181 -5 WHY did physicist defined velocity as displacement divided by time, why not displacement * time? [closed] Nikhil Pant https://physics.stackexchange.com/users/231896 2019-07-12T11:03:49Z 2019-07-12T16:49:46Z <p>V=S/T. As per my knowledge i think ratio as division and it don't give any meaning like this much displacement in this much time. So i think physicists only used division as notion for velocity. But why didn't they used addition multiplication instead like V=S+T or V=S*T or V=T/S.</p> <p>BUT after thinking about it I get to know that multiplication or addition is not a good way to notify velocity as if we use multiplication we have to 1 out of displacement or time to interpret meaning form that velocity and also we r not going to get unique numbers as velocity for totally different conditions like if we say V=10 m/sec then we can interpret it in many ways as 2 meters in 5 sec or 10 m in 1 sec which seems totally wrong</p> <p>But when I take V=T/S then i didn't seen any impropernece in this. So can u tell me why we preferred division as the notation? </p> https://physics.stackexchange.com/q/491130 5 Physicist path integral and cylinder set measures user1620696 https://physics.stackexchange.com/users/21146 2019-07-12T03:13:09Z 2019-07-14T23:50:14Z <h2>Path integral via discretization</h2> <p>So let me start with what seems to be the point of view of physicists (corrections are highly appreciated since this is what I understood!). Let a quantum system with coordinates <span class="math-container">$q_a$</span> and momenta <span class="math-container">$p_b$</span> be given satisfying commutation relations <span class="math-container">$$[q_a,p_b]=i\delta_{ab}.$$</span></p> <p>Further suppose the system has a Hamiltonian <span class="math-container">$H$</span> which is time-independent. Usually in that setting the path integral is introduced as a means to compute the transition amplitude</p> <p><span class="math-container">$$\langle q',t'|q,t\rangle=\langle q'|e^{-iH(t'-t)}|q\rangle=\int\mathfrak{D}x(t) \exp\left\{iS[x(t)]\right\}$$</span></p> <p>This is usually defined by a discretization procedure allied to a Wick rotation to Euclidean time <span class="math-container">$\tau = it$</span> to deal with convergence. The right discretization seems to be derived by slicing the time interval, evaluating <span class="math-container">$\langle q',t'|q,t\rangle$</span> to first order in <span class="math-container">$t'-t$</span>, and imposing some ordering convention. So, for example with a Lagrangian <span class="math-container">$L = T - V$</span> the above integral would be <em>defined</em> as something of the form</p> <p><span class="math-container">$$\int\mathfrak{D}x(t) \exp\left\{iS[x(t)]\right\}=\lim_{N\to \infty} C_N\int \prod_{k=1}^N dx_k \exp \left\{i\sum_{k=1}^N \frac{m}{2}\frac{(x_k-x_{k-1})^2}{\epsilon_N^2}-V(x_k)\right\}\tag{1}$$</span> </p> <p>So: a path integral in Physics is <em>defined</em> by the continuum limit of these aforementioned discretizations.</p> <h2>Cylinder set measures</h2> <p>Now there's the mathematicians point of view on which one studies integration over locally convex vector spaces which are infinite dimensional. In that case, if <span class="math-container">$E$</span> is such a space we perform two definitions:</p> <blockquote> <p><strong>Definition</strong>: Let <span class="math-container">$E$</span> be a locally convex vector space. A <strong>cylinder set</strong> is defined to be a subset <span class="math-container">$C\subset E$</span> of the form <span class="math-container">$$C=\{x\in E : (\ell_1(x),\dots,\ell_n(x))\in C_0\}$$</span> where <span class="math-container">$C_0\subset \mathbb{R}^n$</span> is a Borel subset and <span class="math-container">$\ell_k\in E^\ast$</span> are continuous linear functionals. Equivalently, it is a preimage <span class="math-container">$C = P^{-1}(C_0)$</span> of a Borel set under a continuous linear map <span class="math-container">$P : E\to \mathbb{R}^n$</span>.</p> <p><strong>Definition:</strong> Let <span class="math-container">$E$</span> be a locally convex vector space. A cylinder set measure <span class="math-container">$\nu$</span> is a nonnegative additive set function defined on the <span class="math-container">$\sigma$</span>-algebra generated by cylinder sets of <span class="math-container">$E$</span> such that for any continuous linear <span class="math-container">$P : E\to \mathbb{R}^n$</span> the set function <span class="math-container">$$\nu\circ P^{-1} : B\mapsto \nu(P^{-1}(B))$$</span> is countably additive.</p> </blockquote> <h2>Comparison</h2> <p>If we now compare there are a few points to mention:</p> <ol> <li><p>It seems that cylinder sets capture discretizations. If <span class="math-container">$x$</span> is a continuous path, <span class="math-container">$(\ell_1(x),\dots, \ell_n(x))$</span> is an <span class="math-container">$n$</span>-point discretization. To be even more precise in the case of paths we could take <span class="math-container">$\ell_k(x) = x(t_k)$</span> for some <span class="math-container">$t_1,\dots, t_n$</span> in the interval. In the same way we could take <span class="math-container">$\ell_k(x) = a_k$</span> some Fourier coefficient of <span class="math-container">$x(t)$</span>. I've seem both things done in Physics.</p></li> <li><p>It seems cylinder set measures are in fact a way to define "a measure per discretization". So for each discretization we give a measure - integrate over <span class="math-container">$n$</span> points, integrate over <span class="math-container">$n$</span> Fourier coefficients, so forth.</p></li> </ol> <p>Still, the connection doesn't feel complete for me. The issue is that to define a cylinder set measure we must define <span class="math-container">$\nu$</span> on <strong>the whole algebra generated by cylinder sets</strong>. </p> <p>The Physicist approach seems to do this exactly for a specific collection of cylinder sets. Either for the ones with <span class="math-container">$\ell_k(x) = x(t_k)$</span> or for the ones with <span class="math-container">$\ell(x_k)=a_k$</span> a Fourier coefficient. </p> <p>But there are infinitely many other choices of the <span class="math-container">$\ell_k$</span> which give rise to many more cylinder sets. And one would need still to define <span class="math-container">$\nu$</span> on the <span class="math-container">$\sigma$</span>-algebra itself.</p> <p><strong>The question</strong>: is there really a relation between cylinder set measures and the Physicist discretization of a path integral? If so, how the relation can be made more precise? If not, why not, considering the similarities?</p> <p>For this discussion, please let us consider the Euclidean path integral. So the issue here <em>is not the imaginary exponent</em>.</p> https://physics.stackexchange.com/q/491007 0 Off-shell vs half off-shell vs fully off-shell $T$-matrix tahami https://physics.stackexchange.com/users/192698 2019-07-11T11:23:56Z 2019-07-11T11:34:21Z <p>I know what are on-shell particles, but I want to know what are off-shell, and half off-shell, and fully off-shell states? and how we decide to consider one of these states in evaluating <span class="math-container">$T$</span>-Matrix? </p> https://physics.stackexchange.com/q/490963 -2 Please tell me the difference between electrostatics and electrodymanics and what is the state of equilibrium in both of them [closed] zoha atique https://physics.stackexchange.com/users/236601 2019-07-11T06:04:49Z 2019-07-11T11:11:11Z <p><img src="https://i.stack.imgur.com/yCQq2.jpg" alt="enter image description here"> screen shot of Richard fiegnmnan lectures volume 2 <img src="https://i.stack.imgur.com/mOyq1.jpg" alt="enter image description here"></p> https://physics.stackexchange.com/q/490670 2 Physics Equivalent of IUPAC Gold Book StackUpPhysics https://physics.stackexchange.com/users/204000 2019-07-09T15:23:44Z 2019-07-09T15:23:44Z <p>I wanted to look up a few definitions and found them to vary from source to source so I wondered if there was a book such as IUPAC Gold Book in Chemistry which formally lists and defines almost all phenomenon and principles of physics like angular momentum, spin magnetic moment etc.</p> https://physics.stackexchange.com/q/490203 0 Why definition of potential energy and law of conservation of mechanical energy is misleading several times? Unique https://physics.stackexchange.com/users/230533 2019-07-07T04:23:08Z 2019-07-07T05:18:19Z <p>I regularly see 1 or 2 questions on this website about the definition or application of potential energy.The users fundamentally ask the same thing in every question.</p> <p>What I have learned till now is:-<span class="math-container">$$dU_{system}=-dW_{int,cons}$$</span></p> <p><strong>The change in potential energy of the system is defined as the negative of work done by the internal conservative forces of the system</strong></p> <p>What I have read is that it is defined only for a multi particle system and absolute potential energy is not defined yet.We have only defined relative potential energy.</p> <p>The definition is quite obvious because if we choose a single particle system then no forces would be internal and we can't define potential energy corresponding to it.I have a rigid intuition about the term <strong>negative</strong> in the definition.</p> <p>Why people on this website continuously argue with me that potential energy is also defined for a single particle system?It seems vague to say that <strong>Potential energy of this particle is 10 joules</strong>.</p> <p>Now coming to law of conservation of mechanical energy of a system of particles:-</p> <p>For any system of particles we have from work energy theorem:-<span class="math-container">$$dW_{total}=dK_{system}$$</span> <span class="math-container">$$dW_{int,con}+dW_{int,non-con}+dW_{ext}=dK_{system}$$</span> <span class="math-container">$$-dW_{int,con}=dU_{system}$$</span> <span class="math-container">$$dW_{int,non-con}+dW_{ext}=dU_{system}+dK_{system}$$</span> <span class="math-container">$$dU_{system}+dK_{system}=dE_{mech,system}$$</span> <span class="math-container">$$dE_{mech,system}=dW_{int,non-con}+dW_{ext}$$</span></p> <p>In context of this law,people argue with me about the term of work done by external forces on this system.According to whatever I have learned till now is that this law is also valid for a multi particle system. </p> <p>I want to ask here about the correct and generalized definition of potential energy as well as the correct law of conservation of mechanical energy,if I am wrong?</p> <p>The arguments are in the following question <a href="https://physics.stackexchange.com/questions/488040/question-on-the-definition-of-the-potential-energy-for-a-two-particle-system">Question on the definition of the potential energy for a two particle system</a></p> https://physics.stackexchange.com/q/489827 3 Is four velocity always given by $U^{\mu} = d x^{\mu}/d\tau$? J-J https://physics.stackexchange.com/users/98512 2019-07-04T22:43:35Z 2019-07-05T17:14:21Z <p>I was taught that <a href="https://en.wikipedia.org/wiki/Four-velocity" rel="nofollow noreferrer">four-velocity</a> is defined as <span class="math-container">$${\bf U} = \frac{d \bf x}{d\tau}$$</span> and that it has the components <span class="math-container">$$U^{\mu} = \frac{d x^{\mu}}{d\tau}$$</span> where <span class="math-container">$d\bf x$</span> is the four displacement and <span class="math-container">$\tau$</span> is proper time.</p> <blockquote> <p>My question is simple: is the latter equation (for the components) correct in all coordinate systems?</p> </blockquote> <p>I tried to figure it out the following way:</p> <p><span class="math-container">$${\bf U} = \frac{d}{d\tau}(x^{\mu}{\bf e_{\mu}})$$</span> <span class="math-container">$$= \frac{d x^{\mu}}{d\tau}{\bf e_{\mu}} + x^{\mu}\frac{d{\bf e_{\mu}}}{d\tau}$$</span> <span class="math-container">$$= \frac{d x^{\mu}}{d\tau}{\bf e_{\mu}} + x^{\mu}\frac{d x^{\nu}}{d\tau}\frac{\partial{\bf e_{\mu}}}{\partial x^{\nu}}$$</span> <span class="math-container">$$= \frac{d x^{\mu}}{d\tau}{\bf e_{\mu}} + x^{\mu}\frac{d x^{\nu}}{d\tau}\Gamma_{\nu \mu}^{\alpha}\bf e_{\alpha}$$</span> <span class="math-container">$$= \left(\frac{d x^{\alpha}}{d\tau} + x^{\mu}\frac{d x^{\nu}}{d\tau}\Gamma_{\nu \mu}^{\alpha}\right){\bf e_{\alpha}}$$</span></p> <p>and therefore: <span class="math-container">$$U^{\alpha} = \frac{d x^{\alpha}}{d\tau} + x^{\mu}\frac{d x^{\nu}}{d\tau}\Gamma_{\nu \mu}^{\alpha}.$$</span></p> <p>But the only way this component equation agrees with the earlier one is if <span class="math-container">$$x^{\mu}\frac{d x^{\nu}}{d\tau}\Gamma_{\nu \mu}^{\alpha} = 0$$</span> for all <span class="math-container">$\alpha$</span>. However, I can't seem to prove/disprove this. Obviously if the Christoffel symbols are zero, then it's trivial. But if there are non-zero Christoffel symbols, then is it still zero?</p> https://physics.stackexchange.com/q/489298 -1 Is there a better definition of magnetic field than this? Swami https://physics.stackexchange.com/users/45557 2019-07-02T07:49:14Z 2019-07-02T17:13:42Z <p>It may seem a trivial question but the definition of the magnetic field in everyday books is misleading. "It is the region or area around a magnetic material in which its magnetic force can be felt." It seems magnetic field is a physical area, thus its units must be meter square etc and not Tesla.</p> https://physics.stackexchange.com/q/489094 3 What is irrotational flow? How to judge? enbin zheng https://physics.stackexchange.com/users/176092 2019-07-01T02:36:20Z 2019-07-01T15:37:55Z <p>For example, when the wing moves horizontally, the direction of fluid flow changes first to upward at the leading edge of the wing and then to downward at the trailing edge. Does it rotate? If the direction of motion of the fluid at the trailing edge changes to horizontal, does this also rotate?</p> https://physics.stackexchange.com/q/488536 0 What is a pseudopure state? onurcanbektas https://physics.stackexchange.com/users/99217 2019-06-27T16:09:24Z 2019-06-27T17:41:34Z <p>In the paper titled "Experimental Implementation of the Quantum Baker’s Map" by Weinstein et al. (Phys. Rev. Let. 89 (2002)), the author says something like</p> <blockquote> <p>[...] the pseudopure state corresponding to the state <span class="math-container">$\left |000 \right \rangle$</span>.</p> </blockquote> <p>But, what is a pseudopure state in general ? how it is different from a pure state ? and why do they call the state <span class="math-container">$\left |000 \right \rangle$</span> pseudopure, isn't it a pure state ?</p> https://physics.stackexchange.com/q/488334 3 How can I explain what a kilogram is using Planck's constant? [duplicate] Fullk33 https://physics.stackexchange.com/users/126872 2019-06-26T16:01:32Z 2019-06-27T15:03:34Z <div class="question-status question-originals-of-duplicate"> <p>This question already has an answer here:</p> <ul> <li> <a href="/questions/147433/what-are-the-proposed-realizations-in-the-new-si-for-the-kilogram-ampere-kelvi" dir="ltr">What are the proposed realizations in the New SI for the kilogram, ampere, kelvin and mole?</a> <span class="question-originals-answer-count"> 2 answers </span> </li> </ul> </div> <p>I want to understand what 1 kg <em>represents</em>. For example: I know that 1 second is equal to <span class="math-container">$9\ 192\ 631\ 770$</span> transitions from the microwave radiation that a cesium-133 atom (at <span class="math-container">$0$</span>K) emits, if it's excited just right. I can <em>imagine</em> that. I can see how you would "count" these 9 billion transition until you know, that exactly <span class="math-container">$1$</span> second has passed.</p> <p>Now I would like to know if there is a similar explanation for the kilogram. </p> <p>I understand how Planck's constant has be redefined using methods such as the Kibble balance. I would like to know how I can explain what <span class="math-container">$1$</span>kg is using <span class="math-container">$h$</span>. Here is what I've got so far:</p> <p>Knowing that <span class="math-container">$E=hf$</span> and <span class="math-container">$E=mc^2$</span>, if both of those energies are equal, this gives <span class="math-container">$m=\frac{hf}{c^2}$</span>. So if we want to know what <span class="math-container">$1$</span>kg is, we find the frequency <span class="math-container">$f$</span>, that gives <span class="math-container">$\frac{hf}{c^2}=1$</span>kg, which would be <span class="math-container">$1.3564 \times 10^{50}$</span>Hz. </p> <p>What does this frequency represent? Is it the frequency of light that you would need to "push" an object with a force equivalent to the weight of <span class="math-container">$1$</span>kg? Sorry if my thinking is completely off.</p> <p>Edit: the answer of the question <a href="https://physics.stackexchange.com/questions/147433/what-are-the-proposed-realizations-in-the-new-si-for-the-kilogram-ampere-kelvi">What are the proposed realizations in the New SI for the kilogram, ampere, kelvin and mole?</a> explains in detail how the new units get defined and what their relations are, but does not give a satisfying explanation as to what e.g. a kilogram represents.</p> https://physics.stackexchange.com/q/488218 0 What is a Hamiltonian of a System? Ashwin Balaji https://physics.stackexchange.com/users/193484 2019-06-26T06:40:29Z 2019-06-26T10:35:38Z <p>What is a Hamiltonian of a System? When learning about Hamiltonian for first time it is an object introduced as Legendre Dual Transform of Lagrangian of the same system. And we learn further that it is equivalent to energy of the system. But there are systems where Hamiltonian and Energy doesn't match.(Ex:<a href="https://physics.stackexchange.com/q/11905/">When is the Hamiltonian of a system not equal to its total energy?</a>)</p> <p>We see the use of Hamiltonian in physics is almost everywhere. It may have some deep physical implications about the nature of how things work. So how to understand the Hamiltonian of System other than the Energy concept? ( A more general idea).</p> https://physics.stackexchange.com/q/487982 0 How do I understand Kinetic energy formula? [duplicate] user647077 https://physics.stackexchange.com/users/235415 2019-06-25T06:37:14Z 2019-06-25T09:06:34Z <div class="question-status question-originals-of-duplicate"> <p>This question already has an answer here:</p> <ul> <li> <a href="/questions/535/why-does-kinetic-energy-increase-quadratically-not-linearly-with-speed" dir="ltr">Why does kinetic energy increase quadratically, not linearly, with speed?</a> <span class="question-originals-answer-count"> 18 answers </span> </li> </ul> </div> <p><span class="math-container">$$\frac{mv^2}{2}= Kinetic Energy$$</span> Can you explain me? What is purpose of <span class="math-container">$v^2$</span>, <span class="math-container">$mv^2$</span>, I am trying to understand the formula.</p> https://physics.stackexchange.com/q/487950 1 Klein-Gordon equation propagators: intersection with the support of the source user1620696 https://physics.stackexchange.com/users/21146 2019-06-25T01:02:01Z 2019-06-25T02:03:43Z <p>Let <span class="math-container">$(M,g)$</span> be a globally hyperbolic. Let <span class="math-container">$P = \Box - m^2$</span> be the Klein-Gordon differential operator. Following <a href="http://inspirehep.net/record/1597673/files/lecturenote-3908.pdf?version=1" rel="nofollow noreferrer">Fewster's notes</a>, we may define the retarded/advanced propagators <span class="math-container">$$E^\pm : C^\infty_0(M)\to C^\infty(M)$$</span> as follows. First let <span class="math-container">$f\in C^\infty_0(M)$</span> and consider the problem <span class="math-container">$$P\phi = f$$</span></p> <p>with two conditions imposed on the solution:</p> <ol> <li><span class="math-container">$\operatorname{supp} \phi \subset J^\pm (\operatorname{supp}f)$</span>;</li> <li><span class="math-container">$\operatorname{supp}\phi \cap J^\mp (\operatorname{supp}f)$</span> is compact;</li> </ol> <p>One shows that the problem has unique solution and define <span class="math-container">$E^\pm(f)$</span> to be the solution to the corresponding problem.</p> <p>I'm trying to gain intuition on this. </p> <p>First consider the <span class="math-container">$E^+$</span> case. Condition (1) seems to mean that "to the past of when <span class="math-container">$f$</span> is turned on the solution vanishes". In that sense, it seems it allows us to say that the solution is created by the source <span class="math-container">$f$</span>.</p> <p>Now for the <span class="math-container">$E^-$</span> case. Now condition (1) seems to mean that "to the future of when <span class="math-container">$f$</span> ceases to exist the solution vanishes". In that sense, it seems that the solution is in fact what creates <span class="math-container">$f$</span>.</p> <p>Condition (2), on the other hand, I can't see how to interpret.</p> <p>So, is my intuition on condition (1) correct? What is the intuition for demanding condition (2) when defining <span class="math-container">$E^\pm$</span>?</p> https://physics.stackexchange.com/q/487647 1 How does one obtain $\hbar$ as $\frac{h}{2\pi}$? Anonymous_original https://physics.stackexchange.com/users/232916 2019-06-23T14:30:40Z 2019-06-23T16:13:40Z <p>I'm reading Dirac's Principles of Quantum Mechanics. He defines <span class="math-container">$\hbar$</span> to be the real number satisfying the following relation <span class="math-container">$$uv - vu = i\hbar[u,v]$$</span> where <span class="math-container">$u$</span> and <span class="math-container">$v$</span> are dynamical variables, and <span class="math-container">$[u,v]$</span> is the classical Poisson bracket. He later defines the left hand side of this equation (with the variables replaced with the corresponding operators) to be the quantum Poisson bracket.</p> <p>He then says that from experiments, we must have <span class="math-container">$$\hbar=\frac{h}{2\pi}$$</span> where <span class="math-container">$h$</span> is the constant that was introduced by Planck. How does one get the <span class="math-container">$2\pi$</span>? Is it an approximation? How can we be certain that it is exactly <span class="math-container">$2\pi$</span> to an arbitrary degree of precision? </p> https://physics.stackexchange.com/q/487287 0 What do you mean by Newtonian space? [closed] Aakhyat Singh https://physics.stackexchange.com/users/159899 2019-06-21T12:59:31Z 2019-06-24T04:51:05Z <p>What do you mean by <a href="https://www.google.com/search?as_epq=newtonian+space" rel="nofollow noreferrer">Newtonian space</a>? When you see this question, most of you might be thinking that I am trying to crack a joke or something..but no. This was a genuine doubt which one of my friends raised when we were discussing about NLOM on Whatsapp. He asked this question after I answered his query on what an inertial frame was..I wanted to clarify whether Newtonian space and inertial frame are the same or not.. </p> https://physics.stackexchange.com/q/487124 2 In which sense equations of motion are covariant? SimoBartz https://physics.stackexchange.com/users/206319 2019-06-20T12:56:38Z 2019-06-21T03:30:32Z <p>I read lots questions about what covariance is and I found out that, according to this topic <a href="https://physics.stackexchange.com/questions/230495/lorentz-invariance-of-the-minkowski-metric">Lorentz invariance of the Minkowski metric</a>, we say an object is covariant if it doesn't take the same value on every frame of reference, but the different values are related in a well defined way: the components of a covariant <em>object</em> must satisfy the tensors transformation rule.</p> <p>I understand these definitions but at the same time I heard many times about covariance of an <em>equation</em>. I tried to figure out what is a covariant equation and I noticed that if I have an equality where right and left terms are covariant objects than the equation remains true when I change the frame of reference because both sides transform equally. So i was tempted to say en equation is covariant if it's between covariant objects. On the other hand there are some equations that are said to be covariant but doesn't respect this definition. For example the equations of motion when the frame of reference is changed remains true but they are not made of objects that are covariant.</p> https://physics.stackexchange.com/q/486461 1 What are quasi-regular singularities? user220348 https://physics.stackexchange.com/users/220348 2019-06-16T22:37:56Z 2019-06-17T02:37:11Z <p>The book EXACT SPACE-TIMESIN EINSTEIN’SGENERAL RELATIVITY by Podolsky and Griffiths has a section on Taub-Nut space-time metrics and there is defines the singularity made in the Taub metric as quasi-regular singularity <span class="math-container">$$\mathrm{d} s^{2}=-f(r)\left(\mathrm{d} t+4 l \sin ^{2} \frac{1}{2} \theta \mathrm{d} \phi\right)^{2}+\frac{\mathrm{d} r^{2}}{f(r)}+\left(r^{2}+l^{2}\right)\left(\mathrm{d} \theta^{2}+\sin ^{2} \theta \mathrm{d} \phi^{2}\right)$$</span>s.</p> <p>When we set <span class="math-container">$$\theta=0 \text { and } \theta=\pi$$</span></p> <p>We get a singularity on one of the axises.</p> <p>What is a quasi-regular singularity as opposed to a curvature singularity?</p> https://physics.stackexchange.com/q/486195 0 Definition of closed, compact manifold and topological spaces [migrated] damaihati https://physics.stackexchange.com/users/45429 2019-06-15T11:45:23Z 2019-06-15T11:50:39Z <p>This is a very basic question but I seem not to get a "simple" definition anywhere that is at the same time rigorous and clear. I probably understand basic definitions of topology, topological spaces, open and closed sets, manifolds etc. However, I fail to see what compact or closed topological spaces and manifolds are. </p> <p>I realise that there is a difference between these concepts as applied to topological spaces and manifolds. Also, how do we define the boundary of a topological space and a manifold?</p> <p>I frequently have to encounter these concepts while studying gravity and a clear intuitive picture would help a lot. </p> https://physics.stackexchange.com/q/485037 13 Why is the length of the Kelvin unit of temperature equal to that of the Celsius unit? [duplicate] Thomas https://physics.stackexchange.com/users/141461 2019-06-09T00:08:58Z 2019-06-10T09:22:20Z <div class="question-status question-originals-of-duplicate"> <p>This question already has an answer here:</p> <ul> <li> <a href="/questions/65857/why-is-a-degree-celsius-exactly-the-same-as-a-kelvin" dir="ltr">Why is a degree Celsius exactly the same as a Kelvin?</a> <span class="question-originals-answer-count"> 2 answers </span> </li> </ul> </div> <p>The Celsius unit is arbitrarily defined, based on the boiling and freezing point of water. Is it a coincidence, then, that the SI unit of temperature Kelvin, which is used in all natural equations, has the same length as the Celsius unit?</p> https://physics.stackexchange.com/q/484533 0 How do we define "mass" in the context of particle physics and relativity? James Goetz https://physics.stackexchange.com/users/129342 2019-06-06T03:54:08Z 2019-06-07T06:50:18Z <p>In laypersons terminology, <a href="https://en.wikipedia.org/wiki/Mass" rel="nofollow noreferrer">mass</a> is defined as the amount of matter. However, consider the following:</p> <ol> <li>The <span class="math-container">$W$</span> and <span class="math-container">$Z$</span> bosons have mass.</li> <li>An antiparticle has the same mass as its corresponding particle.</li> </ol> <p>Also, the mass of particles is typically defined by fractions of a kilogram, but a kilogram on a scale is a measure of weight that is relative to nearby gravity.</p> <p>Please help me. I am a layperson in physics while I want to define the mass of a physical object while considering all of the above.</p> https://physics.stackexchange.com/q/484297 2 How can tempered distributions be paths? user1620696 https://physics.stackexchange.com/users/21146 2019-06-04T16:56:54Z 2019-06-04T20:08:58Z <p>I'm reading the Appendix A of Glimm and Jaffe book "Quantum Physics: a functional integral point of view", and there is something that I'm missing</p> <p>In section A.4 the authors talk in a very general context about functional integration. If I got it right they are considering a sequence of Hilbert spaces <span class="math-container">$\mathscr{H}_n$</span> and setting <span class="math-container">$$\mathscr{H}_\infty=\bigcap_{n\in \mathbb{Z}}\mathscr{H}_n,\quad \mathscr{H}_{-\infty}=\bigcup_{n\in \mathbb{Z}}\mathscr{H}_n$$</span></p> <p>They say <span class="math-container">$\mathscr{H}_\infty$</span> is a nuclear space and <span class="math-container">$\mathscr{H}_{-\infty}$</span> its dual. They exemplify with <span class="math-container">$\mathscr{S}$</span> the Schwartz space and <span class="math-container">$\mathscr{S}'$</span> the corresponding distributions.</p> <p>Then the authors set out to study measures and integration on <span class="math-container">$\mathscr{H}_{-\infty}$</span>:</p> <blockquote> <p>We take Gaussian measures as the starting point for integration over infinite dimensional spaces. Other, non-Gaussian, measures are then obtained by perturbation, e.g., through the Feynman-Kac formula. The dual of a nuclear space (i.e. <span class="math-container">$\mathscr{H}_{-\infty}$</span>) provides a convenient framework for studying Gaussian measures over infinite dimensional spaces.</p> </blockquote> <p>I'm really missing the point of considering this kind of Hilbert spaces, specially these ones defined by these sequences.</p> <p>Further on the authors even call an element of <span class="math-container">$\mathscr{S}'$</span> a path. How can that be? A path is a mapping <span class="math-container">$\gamma : [a,b]\to \mathbb{R}^d$</span> and an element of <span class="math-container">$\mathscr{S}'$</span> is a map <span class="math-container">$\varphi : \mathscr{S}\to \mathbb{R}$</span> acting on functions<span class="math-container">$f : \mathbb{R}^d\to \mathbb{R}$</span>. I can't see why an element of that space is a path!</p> <p>Also I never thought that the space of paths needed to carry any inner product structure. For instance, I always considered that the relevant space for non-relativistic quantum mechanics was <span class="math-container">$C^0([a,b];\mathbb{R}^d)$</span>.</p> <p>So what is the intuition here? Why consider Hilbert spaces - and hence an inner product structure - as the spaces of functions one is integrating over? Furthermore, why <em>nuclear</em> spaces?</p> https://physics.stackexchange.com/q/483955 1 Scattering amplitudes vs correlators AoZora https://physics.stackexchange.com/users/226311 2019-06-02T23:03:14Z 2019-06-04T09:54:53Z <p>What are the practical differences between correlators and scattering amplitudes in quantum field theory?</p> <p>On a very practical level: scattering amplitudes describe the evolution of an IN state into an OUT state; how the dynamics is encoded in correlators instead?</p> <p>I think that one difference in particular is that in a conformal field theory (and therefore also at a fixed RG point of a QFT) scattering amplitudes, introducing external momenta (i.e. scales), seem to break the conformal symmetry, while correlators don't. Is this a problem for scattering amplitudes? Should this be a reason to prefer correlators in a CFT regime?</p> <hr> <p>On a somewhat more technical level:</p> <p>there are many advanced mathematical results for scattering amplitudes such as recursive equations and algebraic relations coming from polylogs or the general structure of the feynman diagrams, or even more strongly the S-matrix bootstrap (amplitudes are completely determined once asked analycity, Poincarè invariance, locality and unitarity); can these be translated to correlators or are these objects in some way less manageable? If correlator-oriented versions of these properties are possible, how comes that in mathematical physics there is much more focus on scattering amplitudes? (maybe it's just my impression?)</p> <p>Don't worry if you cannot address this last point in your answer, maybe it would require too much effort.</p> https://physics.stackexchange.com/q/483631 1 Questions about an inertial frame rsnelsonjose https://physics.stackexchange.com/users/210660 2019-06-01T01:40:01Z 2019-06-01T07:22:53Z <ol> <li>Can someone explain to me what I put in bold? </li> </ol> <p>Inertial frame definition:</p> <p>When the coordinate axes are stationary <strong>with respect to the mean position of the "fixed" stars</strong> or if they move with uniform linear velocity, without rotation, with respect to the stars.</p> <ol start="2"> <li>When the author says the following (see bold text), what makes this frame "slightly not inertial"?</li> </ol> <p>Of course, it should be noted that measurements made with great precision could show that <strong>the frame of "fixed stars" is slightly not inertial</strong>.</p> <ol start="3"> <li>The reason why a frame of reference on earth, both on the surface and at its center, is not inertial is because of its acceleration of circular motion? Doesn't it matter that the angular velocity of the earth is constant 360°/24h (daily rotation) and 360°/365days (annual rotation)?</li> </ol>