Noether's Theorem

6 Key excerpts on "Noether's Theorem"

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  The Standard Model in a Nutshell

  • Dave Goldberg(Author)
  • 2017(Publication Date)
  • Princeton University Press

    (Publisher)

  3 Noether’s Theorem Figure 3.1. Emmy Noether (1882–1935), c. 1910. Noether developed the general connection between symmetry and conservation laws. Conservation laws are the bread and butter of physics. Quantities like angular or linear momentum or electric charge are useful because you can measure them at one moment, and so they will remain forever. For much of the history of physics, conservation laws were taken to be almost axiomatic facts of nature. Galileo argued for something very much like the conservation of momentum in his Two New Sciences [72] in 1638, which ultimately gave rise to Newton’s first law of motion. Benjamin Franklin, in a letter to Peter Collinson in 1747 [67] noted a similar effect for what we now call electrical charges: the Equality is never destroyed, the Fire only circulating. Hence have arisen some new Terms among us. We say B (and other Bodies alike circumstanced) are electrised positively; A negatively : Or rather B is electrised plus and A minus . Energy conservation, too, seems experimentally to be a fact of nature, a property noted by Galileo in the motion of pendulums, and culminating, in more modern language, in the first law of thermodynamics as elucidated by Rudolf Clausius in 1850 [39]. Conservation laws are useful to be sure, but by the turn of the twentieth century, no one had any real idea of why nature conserved some quantities and not others. The connection 44 | Chapter 3 Noether’s Theorem had to wait until 1918, when the mathematician Amalie “Emmy” Noether published her eponymous theorem [117, 115] relating conservation laws to symmetries. We’ve already seen how useful symmetries can be by construction of only Lorentz-invariant Lagrangians. However, Noether’s theorem demonstrated that the seemingly “obvious” symmetries of nature—invariance over time and throughout space, for instance—can have profound implications and predict (or really, retrodict, given the order of scientific discovery) important conservation laws.

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  The Philosophy and Physics of Noether's Theorems

  A Centenary Volume

  • James Read, Nicholas J. Teh(Authors)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  She has by herself completely set out in proper form the mathematical ideas that I use in connection with the physics problems related to [the action], which will be presented in a note to appear shortly in these Nachrichten. Thus by 1918, Noether’s theorem was, in the eyes of Hilbert, Klein, and Noether, the proof that there is no proper conservation law in general relativity: which Klein interpreted – 26 The above summary of the argument in Klein (1918a, p. 504) and Einstein (1918; 1918a, p. 513) follows Brading (2005, p. 126), which indeed seems to me a correct interpretation both of Klein’s letter and of Einstein (1918b, p. 513) reply. 27 Noether’s use of the word ‘subgroup’ here appears to be stronger than the modern use. Her statement assumes not just the group of translations being a subgroup of an infinite group, but also its being embedded in a specific way: see Section 9.2.3. I thank Bartlomiej Czech for pushing me on this point. 28 Klein (1918b, p. 189). 9 Noether’s Theorems and Energy in General Relativity 205 with Hilbert’s and Noether’s approval, one should add – as the energy equations’ lacking physical content. I will return to Einstein’s proposal for a conservation law in general relativity, and its problems, in more detail in Section 9.1.3. In the next section, I first illustrate Noether’s two theorems in a well-known example. 9.1.2 Illustrating Noether’s Theorems in the Maxwell Theory In this paper, I will use the Maxwell theory of electromagnetism as a running example to illustrate various aspects of Noether’s theorems, since this theory already exemplifies some of the issues that appear in general relativity. 29 Thus, let me recall how the theorems work for elementary electromagnetism without sources: take the following action, defined on Minkowski spacetime: S [A] =  d 4 x L(A; x ) = − 1 4  d 4 x F μν F μν F μν := ∂ μ A ν − ∂ ν A μ .

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  Symmetries of Nature

  A Handbook for Philosophy of Nature and Science

  • Klaus Mainzer(Author)
  • 2013(Publication Date)
  • De Gruyter

    (Publisher)

  But more about that later, when we discuss the above-referenced theories. In addition to her significant results in mathematical physics, E. Noether (as well as E. Artin, Β. L. van der Waerden et al.) made major contribu-tions to the foundation of modern algebra. And last but not least, it should be noted that Emmy Noether is among the - regrettably few - great female mathematicians in the history of mathematics. In the 20th Century, she not only had to contend with the general discrimination against women in sci-ence, but also with racial persecution in the Germany of her era. The fact that, even amid the misery of her personal life, she was able to make funda-mental contributions to mathematical symmetry in the Platonic tradition of ideal beauty and harmony is part of the mysterious dialectic of the history of science. 130 3.33 Extremal Principles and the Pre-established Harmony of Nature In Noether's Theorem, the existence of conservation laws is concluded from invariance characteristics of Hamilton's action integral. As the title of Noether's work (Invariant Variational Problems) indicates, the conserva-tion laws are therefore traced to the symmetry characteristics of a variational problem (namely of Hamilton's action integral). This is historically the old-est root of Noether's Theorem, which goes back to Leibniz. It is the history of variational theory and Leibniz's principle of least action, which is close-ly connected to the modern philosophy of nature. Since Leibniz's introduc-tion of this principle, it has been a focal point of speculation concerning a universal law of nature which would not only explain physical events but could even prove the existence of God and serve as an expression of a pre-established harmony of nature. Although such physical-theological consid-128 See E. Noether's 2nd Theorem in her work entitled Invariante Variationsprobleme (See Note 124); see also P.

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  Gravitation

  • A R Prasanna(Author)
  • 2016(Publication Date)
  • CRC Press

    (Publisher)

  As conservation laws are fundamental for understanding nature, Noether’s theorem is considered as the most profound feature of Lagrangian mechanics. The theorem states that ‘every differential symmetry of the action function of any physical system has a corresponding conservation law’ One can find an excellent review of Noether’s theorem and its applications, for relating continuous symmetries of a system with conserved quantities of the system, in [ 5 ], section 12.7. 2.1.6.1  Effective potential Before ending this section, it is useful to introduce an important concept in the context of central force problems–the effective potential, which is a very useful tool in determining the nature of the particle orbits around the central body, by just knowing the particle energy and angular momentum, along with its initial position. If the system is stationary (time independent) and axisymmetric, and if the force applied is conservative and central (depends only on the radial coordinate), then F → (r) = − r → ∂ Φ (r) ∂ r, (2.39) with r 2 θ ˙ = h and the energy E being constants of motion. One can then see that in any three–dimensional central potential, the total mechanical energy E can be written as E = m r ˙ 2 2 + L 2 2 m r 2 + Φ (r) (2.40) with L = m h, which is similar in form to a one-dimensional motion in a potential V (r) = L 2 2 m r 2 + Φ (r) ; E = m r ˙ 2 2 + V (r). (2.41) The potential V(r) is thus called an effective potential, V eff and L 2 2 m r 2, the centrifugal potential. As an example, if one considers the Newtonian gravitational potential,. Φ(r) = – GM / r, then just the plot of (V (r) = L 2 2 m r 2 − G M m r) can reveal a good amount of information about the particle orbits. From the plot in figure 2.6, one can see that while at large r, the force is attractive, at small r, the force is repulsive. The minimum of the potential, occurring at r = L 2 G M m 2 corresponds to the equilibrium point, where the particle has a stable circular orbit

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  Effective Field Theories

  From the Ionian School to the Higgs Boson and Beyond

  • Alexey A Petrov, Andrew E Blechman(Authors)
  • 2015(Publication Date)
  • WSPC

    (Publisher)

  Chapter 2

  Symmetries

  2.1Introduction

  The most vital part of effective field theory is knowing what symmetries apply to your system. This knowledge can get you very far in describing the nature of the problem, constructing model-independent equations to describe dynamics, and put constraints on matrix elements. This chapter will be a review of some of the more important results that follow from symmetries. We will discuss Noether’s and Goldstone’s theorem, and the consequences that arise from them. We will discuss various examples, but for simplicity we will stick mostly with scalar fields wherever we can, avoiding fermions until we need them for anomalies.

  2.2Noether’s Theorem

  The chief reason why symmetries are important is due to a theorem in Lagrangian mechanics known as Noether’s Theorem :

  Theorem 2.1 (Noether). Every continuous symmetry of the action (and path integral measure) implies a conservation law .

  The caveat about the measure being invariant is important to handle the possibility of quantum anomalies, as we will see at the end of this chapter.

  Proof . Consider a Quantum Field Theory (QFT) with fields ϕ a and action S [ϕ ] that is presumed to be invariant under a global transformation of the form:

  where ∈ is an infinitesimal parameter. Although the action is only supposed to be invariant under these transformations for constant ∈ , let us consider the behavior of the action under the above field transformation when ∊ is allowed to vary with space-time:

  The last term is there regardless of whether ∈ is constant or a function of space-time, so if S [ϕ ] is to be invariant under the global transformation in Eq. (2.1) this term must take the form The quantity will generally be nonzero if Δϕ

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  Discover Physics

  • Benjamin Crowell(Author)
  • 2018(Publication Date)
  • Studium Publishing

    (Publisher)

  2 These forms of energy can be converted into others, such as the energy your car has when it’s moving, the light from a lamp, or the body heat that we mammals must continuously produce. We’ll first develop a real scientific definition of energy, and then relate it to symmetry in section 4.4.

  Kinetic energy

  Symmetry arguments led us to the conclusion that an isolated object or ray of light can never slow down, change direction or disappear entirely. But that falls short of being a conservation law. A full-fledged conservation law says that even when we have many objects interacting, the total amount of something stays constant. Is there any reason to believe that energy is conserved in general? The planet earth, c, is a large, complex system consisting of a huge number of atoms. It keeps on spinning without slowing down, which is evidence in favor of energy conservation. What about the spinning coin in figure d, however? Does its energy disappear gradually?

  Scientists would have thought so until the nineteenth century, when physicist James Joule (1818-1889) had an important insight. Joule was the wealthy heir to a Scottish brewery, and funded his own scientific research. As an industrialist, he had a practical interest in replacing steam engines with electric ones that would be more efficient, and cost less money to run. Scientists already knew that friction would cause a spinning coin to slow down, and that friction made engines less efficient. They also knew that friction heated things up, as when you rub your hands together on a cold day. Joule, however, realized that it went deeper than this: there was a conserved quantity, which ended up being called energy. When we first start the coin spinning, its energy is in the form of motion, with its atoms all going in circles. As it slows down, the energy isn’t disappearing, it’s being converted into another form: heat. We now know that heat is the random motion of atoms. As the coin rubs against the ground, the atoms in the two surfaces bump into each other, and the amount of random atomic motion increases. The organized motion of the atoms in the spinning coin is being converted into a disorganized form of motion, heat.

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