Review and Notation (Special relativity)

Transcription

1 Review and Notation (Special relativity) December 30, :35 PM Special Relativity: i) The principle of special relativity: The laws of physics must be the same in any inertial reference frame. In particular, signal propagation in all such frames has a limiting velocity of c= m/s. ii) Any reference frame (coordinate system) moving with a constant velocity relative to an inertial reference frame is itself inertial. iii) All equations of motion must be Lorentz Covariant (i.e. their form is unchanged by a Lorentz transformation) (1). iv) "Natural" units: We ll take mass (M), action (A), and velocity (V) as the fundamental dimensional quantities (A has units of angular momentum). All other quantities can be expressed in terms of these: We ll then choose the units of A and V, such that all quantities can be expressed in terms of dimensions of energy: With velocities specified in units of c and action specified in units of ħ. Thus: M. Gericke Phys 7560, Relativistic QM 1

2 January 6, :58 PM 1) Lorentz Transformations: a. A Lorentz transformation relates event coordinates (t,x) used by one inertial observer to the coordinates (t,x ) used by another inertial observer who is moving with velocity v relative to the first observer. A Lorentz transformation is defined as a coordinate transformation that leaves the interval S 12 2 between events x 1 and x 2 invariant: This is a necessary condition (1). Null intervals (S 12 2 = 0 ) represent two events that can be connected with a light signal. Invariance of S 12 2 implies that all inertial observers agree on the speed of light (c = 1). b. Four-vector notation (conventions as in J.D. Jackson): Since space and time are coupled for relativistic particles, it is convenient to introduce the following new notation: So (1) The invariance of S 12 2 is a necessary condition on the Lorentz transformation, but it is not sufficient by itself, to define the transformation. If we assume however, that space-time is homogeneous and isotropic, then the transformation must be linear, and this uniquely defines the form of the transformation (see Jackson ch. 11.2). M. Gericke Phys 7560, Relativistic QM 2

3 January 8, :25 PM We also define the metric tensor Which allows us to define Here, we have introduced a summation notation, for repeated contravariant and covariant Greek indices; the sum goes from 0 to 3. Note that (1) This is the form of the Kronecker Delta used in relativistic calculations. With this, we can write: (1) Note that g and g are inverses of each other: g g = g =. This holds because we are using Cartesian coordinates, but may not hold in other coordinate systems. M. Gericke Phys 7560, Relativistic QM 3

4 Review and Notation (Special Relativity) January 8, :43 PM c. Lorentz transformations are rotations, boosts and translations, in this, so called, 4-dimensional Minkowski space, that keep the total interval S122 unchanged: Translations clearly leave S122 invariant, since they drop out. We then claim that the interval S122 remains invariant if or, in matrix notation: Notation: Note that a ' takes contravariant ' and covariant ', while a ' takes covariant ' and contravariant '. The positioning here is important, since we must keep track of two things: Thus: M. Gericke Phys 7560, Relativistic QM 4

5 January 8, :18 PM 2. Lifetimes In the rest frame of a particle, x = 0, t = ( = proper time ) In the laboratory z = vt (x = y = 0) x ' x ' = 2 = x x = t 2 (vt) 2 Life times are longer when measured in the lab ( time dilation ). 3. Four-Vectors Four vectors are defined by how they transform under homogeneous Lorentz transformations: If then is a contravariant four-vector is a covariant four vector Examples: i. interval with respect to the origin ii. four-vector derivative iii. velocity four-vector iv. momentum four-vector v. current four-vector with charge and current densities M. Gericke Phys 7560, Relativistic QM 5

6 January 8, :15 PM To check this we start with an interval in the primed reference frame and transform it back to the unprimed interval: It also follows that Which means that a -1 exists (as is physically obvious because the anti rotation has to exists if the rotation exists) and is given by Thus, homogeneous (meaning ) Lorentz transformations are pseudoorthogonal (1) (orhtogonal would mean that a -1 = a T ). (1) The matrix elements of a must be real to ensure that all coordinates are real. M. Gericke Phys 7560, Relativistic QM 6

7 Review and Notation (Special Relativity) January 8, :35 PM Nomenclature: Only proper or orthochronous (also called restricted ) transformations can be constructed by compounding infinitesimal boosts and rotations. Other classes include discrete transformations like parity and time-reversal. Note that a depends on the relative velocity (v) and relative orientation of the two frames, but we ll almost never need an explicit form. d. Example: Lorentz boost along the z direction Take (definition of the origin; Transverse components are unaffected : when the clocks start) So only the z and t components mix, and we can express the transformation as a pseudo rotation that depends on one parameter: This parameterization satisfies eqn. 1.3: M. Gericke Phys 7560, Relativistic QM 7

8 January 8, :56 PM To identify the parameter, note that if z = 0, z = vt. This is just the location of the origin of the primed coordinate system. Thus from eqn. 1.4, we find (1) We then have e. Since Lorentz transformations preserve the interval x 2 = t 2 x 2, then by considering all inertial frames with common origin, we can construct a family of Lorentz-invariant surfaces, each with a given value of x 2. These curves are hyperboloids with asymptotes at x = t. We illustrate this here for two dimensions where the surfaces become hyperbolas. (1) The sign of is conventional. The result for that a ' is the same if we let = - in the intermediate steps. The signs of the diagonal follow from the requirement that a ' ' for v 0. M. Gericke Phys 7560, Relativistic QM 8

9 January 8, :07 PM Definitions: Space-like: One can always find an inertial frame where t = 0 and z 0, which means that the event and the origin have the same time component (the event and the origin do in fact not coincide, but things happen and are detected simultaneously). Two events with a spacelike interval can never communicate via a normal signal, since z t. Time-like: One can always find a frame where t 0 and z =0, so that the event occurs at the origin, but at a different time. All points in the forward light cone (see figure I.2) have t z and hence can be reached by a signal leaving the origin. This relationship is not changed by any proper or orthochronous transformation. we can separate the light cone into a future and a past. Moreover, points that can be connected with a signal stand in a causal relationship that cannot be changed by any Lorentz transformation. M. Gericke Phys 7560, Relativistic QM 9

10 January 8, :31 PM We can prove that (ii) is a four vector using the transformation law for x using the chain rule: So M. Gericke Phys 7560, Relativistic QM 10

11 January 12, :53 AM Any scalar product of a contravariant and a covariant four-vector is a Lorentz invariant: 4. Tensors a. Tensors are also defined by their properties under homogeneous Lorentz transformations : is a contravariant tensor rank 2,3,4,... b. The metric tensor This follows from the definition of the Lorentz transformation (eqn. 1.3) g is a rank-2 tensor (1). It s easy to verify that g has the same form in all frames. 1. From eqn. 1.3: g = a ' a ' g ' ' a a g = a a ' a a ' g ' ' = g (having used the inverse expressions from earlier). So a a g = g (re-label) a ' a ' g = g ' '. M. Gericke Phys 7560, Relativistic QM 11

12 January 13, :36 PM c. The anti-symmetric symbol: Convention: This is the analogue to the Levi-Civita symbol for 3 dimensions. If we define to have the same form in all frames, how does it transform? Based on the properties just stated above, we can write the following: This is a pseudo tensor of rank 4 (1). It is a pseudo tensor, because det(a) = 1, where +1 means that we have a restricted Lorentz transformation. 1. Thus, for all observers connected by restricted Lorentz transformations takes the same form. However, if we change from a right-handed to a left-handed coordinates system (an improper transformation with det(a) = -1), all entries in change sign. This is as expected for a tensor that tells us how to make cross-products. M. Gericke Phys 7560, Relativistic QM 12

13 Review and Notation (Electromagnetism) January 13, :38 PM d. Example: The Electromagnetic Field Tensor: one can see that is antisymmetric. We could prove that this is a second-rank tensor, using the (messy) transformation laws on E and B, but we ll use an alternative approach, showing that Maxwell s equations follow from: and then using the transformation properties of the other pieces in these equations. Here we have and we use natural units for sources in the vacuum: M. Gericke Phys 7560, Relativistic QM 13

14 Review and Notation (Electromagnetism) January 13, :51 PM From the first equation in eqn we get From the second equation (recall that = -1 for odd permutations of 0,1,2,3) we get M. Gericke Phys 7560, Relativistic QM 14

15 Review and Notation (Electromagnetism) January 13, :58 PM Since we know that Maxwell s equations hold in any inertial reference frame, and we know the transformation properties of and, we can say immediately that is a second rank tensor(1,2). Note that we can satisfy the homogenous Maxwell equations automatically by defining where is the four-vector potential. Since then simply by symmetry. We then learn immediately how to express the E and B fields in terms of the potentials: 1. This result follows from the inhomogeneous Maxwell equations: The index must transform as a four-vector to match with j ; since F is antisymmetric, index must also transform as a four vector, which thus makes a scalar, when acted on with. The homogeneous equation is less useful (since the RHS is zero), and we include it here just to complete Maxwell s equations. 2. If we define a dual tensor then the homogeneous equation reads simply : and M. Gericke Phys 7560, Relativistic QM 15

16 Review and Notation (Quantum Mechanics) January 15, :32 AM (Very) Quick Review of Quantum Mechanics A physical system is described by a state vector (t) in an abstract Hilbert space. The state vector allows one to predict the statistical outcome of any experiment. It thus describes the system as completely as possible. Observables are represented by linear hermitian operators. The linearity of the operators is important because with that, the observable of a system can be obtained from a superposition of states. 3. Poisson bracket -i commutator (1) 4. Schrödinger equation determines dynamics linear in (superposition can be used) linear in / t 5. Continuity equation on probability flux allows a point wise interpretation of theory (1) Together with our definition of observables this quantization procedure ensures the correct classical limit (Ehrenfest s theorem) M. Gericke Phys 7560, Relativistic QM 16