Erweiterte Suche der Leibniz Universität Hannover

a) Show that s+ t+ u = m2A +m 2 B +m 2 C +m 2 D. b) Find the centre-of-mass energy of A, in terms of s, t, u, and the masses. Then (V , [ , ]) is a Lie algebra. c) The exponential map allows to map a Lie algebra element X into an group element exp(X) of the corresponding Lie group. b) Show that the covariant derivative Dµφ := (∂µ − igAµ)φ transforms covariantly under (8), i.e.

Problem 6: spontaneous symmetry breaking II — Goldstone theorem For the complex scalar field φ = (ϕ1 + iϕ2)/ √ 2 the Lagrangian takes the form L = (∂σφ) ∗(∂σφ) − µ2φ∗φ− λ(φ∗φ)2 , (2) which has a continuous global U(1) ∼= SO(2) symmetry given by φ(x) 7→ exp(iχ)φ(x). What are the mass terms for the two real scalar fields η(x) and ρ(x)? 1 of 2 Olaf Lechtenfeld Marcus Sperling Fundamental Interactions Tutorial #3 Thursday, May 12 Problem 7: sponta...

Thus, decompose 2 ⊗ 3 and 2 ⊗ 1 into irreducible isospin multiplets and express the states in triple products of u and d. c) Investigate the symmetry properties of the states in the isospin quadruplet 4 and the two isospin doublets under the exchange qi ↔ qj for q1q2q3 where the qk ∈ {u,d}. d) What is the symmetry property of the productΦflavour χspin under exchange of any two quarks? What is the spin wave function for a baryon made up of u an...

(5) c) Prove that u(p) and v(p) from (4a) are indeed solutions of the Dirac equation (1). d) Suppose ~p = (0, 0,p), confirm that the solutions can be re-written as follows: u(1)(p) =   √ E+m 0√ E−m 0   , u(2)(p) =   0√ E+m 0 − √ E−m   , (6a) v(1)(p) =   0 − √ E−m 0√ E+m   , v(2)(p) =   √ E−m 0√ E+m 0   . g) Show that the Hamilton operator of (1) takes the form H = ( m · 12 ~σ · ~p ~σ · ~p −m · 12 ) . (8) h) Prov...

Olaf Lechtenfeld Marcus Sperling Fundamental Interactions Tutorial #4 Thursday, June 2 Problem 8: Two-particle decay For the decay 1 → 2 + 3, where particle 1 is assumed to be at rest, the decay rate is given by Γ = S 32π2m1 ∫ |M|2 δ4(p1 − p2 − p3)√ ~p22 +m 2 2 √ ~p23 +m 2 3 d3~p2 d 3~p3 , (1) where mi is the mass of the ith particle, pi is its four-momentum, and ~pi is its spatial momentum. Problem 9: Z width We recall that the Standard Model...

(i) A Dirac monopole of magnetic charge g and a point particle with electric charge q remain motion- less at a distance of 2a. (i) Generalise the quantisation condition from Problem 14(iii) to the scenario of two dyons (q1,g1) and (q2,g2). (iii) A CP transformation acts on the charges via (q,g) 7→ (−q,g). Problem 16: charged particle in a magnetic monopole field The Newton’s equation of motion for an electrically charged point particle (mass m...

In this problem, we consider neutral kaons K0 = ds̄ and K̄0 = d̄s. d) Suppose an experiment starts with a pure beam of K0 states at time t = 0, i.e. The wave function ψ evolves as |ψ(t)〉 = 1 √ 2 ( |KS〉 · e−imSt− 1 2ΓSt + |KL〉 · e−imLt− 1 2ΓLt ) (6) Compute the decay rate Γ(K0|t=0 → 2π0) (as a function of time) for the state ψ, which was produced as K0 at t = 0, into two neutral pions. In summary, note that the neutral kaons are typically produ...

[H2] Lagrangians in Classical Field Theory [2 pts] Repeat the reasoning of [H1] for the case of a local covariant field theory with Lagrangian density L(φ, ∂µφ), i.e. show that the equations of motion are invariant under addition of a total divergence L 7→ L′ ≡ L+ ∂µfµ(φ) in the two ways mentioned above. Show that in this manner the same equations of motion as in (b) are found. Fix the sign of the parameter b with the help of the requirement t...

This is the one important algebra in two-dimensional conformal field theory, since it is the algebra of the generators of arbitrary locally conformal (i.e. In order to further analyse the nature of a two-dimensional conformally invariant quantum field theory, one has to know how physically relevant representations of the algebra Vir on a space of states look like. [P3] Gradation Show that the state L−n|h〉 has weight h + n. Convince yourself th...

This periodicity implies that the field Φi has a Fourier expansion of the form Φi(x0, x1) = ∞∑ n=−∞ einx 1 f in(x 0) , where the functions f in(x 0) are, at this state, arbitrary. [P1] The mode expansion of the free field To further restrict the functions f in(x 0) proceed in the following way: (a) Compute the equations of motion and the canonical conjugate momenta of the fields Φi. You should again distinguish the cases n 6= 0 and n = 0. for ...

June 2009 FREE FIELD REALIZATION AND VERTEX OPERATORS We learned in the lecture that in conformal field theory, fields are divided into two classes, the primary fields Φh(z, z̄), and their descendant fields. We know one exception so far: Considering a theory of a free massless scalar Boson φ(z, z̄), we saw that the current J(z) = i∂zφ(z, z̄) is a chiral primary field of weight h = 1. [P2] Operator algebra Compute the leading term of the OPE of...

Hence, in the conformally invariant quantum field theory of scalar Bosons, the quantum energy-momentum tensor reads T (z) = − 1 2 ∑ j :∂Φj(z)∂Φj(z): = − 1 2 ∑ j lim w→z [ R∂Φj(w)∂Φj(z) + δjj (z − w)2 ] , where we now consider a theory of various species of free Bosons. To be specific, we call the number of Boson species c. To solve the following exercises, all you need is the operator product expansion (OPE) RΦj(z, z̄)Φk(w, w̄) = −δjk [log(z −...

φN (wN )|0〉 for all ε. This means that the variation of the correlation function works as a derivation, involving the variations of the individual fields. Since the above equation is valid for all ε, we can use it together with the OPE of the energy momentum tensor with primary fields to find the conformal Ward identity 〈0|T (z)φ1(w1) . Therefore, the action of the energy momentum tensor on a correlation function of primary fields is given in ...

In this tutorial, we discuss one very important example of a conformally invariant theory, the free massless scalar Boson. (b) Compute δ √ det g by using the relation √ det g = exp(1 2 log(det g)). [P2] Energy-momentum tensor In the preceding exercise, you should have obtained the energy-momentum tensor for the theory of the free massless scalar Boson to read Tµν = −∂µφ∂νφ+ 1 2 gµνg ρσ∂ρφ∂σφ . (c) Compute the trace of the energy-momentum tensor.

Verschiebung des Klausurtermins Große Übung Öffentliches Recht 3.Klausur 07/01/2015 Neuer Schreibtermin: Freitag, 10.07.2015 von 8.30 11.30 Uhr Raum 1507.201 Aufgrund der Wetterprognose, die für Freitag, den 03. Juli 2015 Spitzentemperaturen von bis zu 37 Grad für den Großraum Hannover vorhersagt, wird die 3. Klausur in der Übung für Öffentliches Recht bei Herrn Professor Butzer nicht am Freitag 16 19 Uhr geschrieben werden. ...