Название: Foundations of Quantum Field Theory
Автор: Klaus D Rothe
Издательство: Ingram
Жанр: Физика
Серия: World Scientific Lecture Notes In Physics
isbn: 9789811221941
isbn:
In order for
In the non-relativistic limit
so that we recover the Schrödinger 1-particle wave function of the non-relativistic theory. Since the argument of the exponential appearing in the Fourier integral is not Galilei-invariant, the corresponding non-relativistic field ϕnr(
3.2Difficulties with the wave equation
As we now show, the wave equation (3.1) presents a number of problems which will eventually lead us to abandon it.
Locality
The wave equation (3.1) has an unwanted property: In order to determine the change in the solution at the point
which upon using the Taylor expansion in p2 becomes
Probability interpretation
In order for the wave function to have the interpretation of a probability amplitude, it should have at least the following properties:
(i) Its normalization with respect to some suitable integration measure should be independent of time, as well as of the choice of inertial frame.
(ii) The associated probability density should be positive semi-definite.
The second condition is satisfied by defining the probability density by
Furthermore, adopting the following definition for the scalar product
and making use of the equation of motion (3.1), one finds that the normalization of the wave function is preserved in time:
The probability of finding a particle anywhere in space should, however, also be independent of the choice of inertial frame. Since the wave function ϕ(
For the case of a free particle this defect is easily repaired by adopting a new definition for the scalar product:
Indeed, making use of the Fourier decomposition (3.4) for a free particle wave function, one computes
Because of the invariance property (2.43) of the integration measure, and the scalar property (3.5) of the Fourier amplitude, we conclude that the normalization of the wave function is a Lorentz invariant with respect to the scalar product (3.6). In fact, one easily shows that it has all the properties expected from a scalar product:
and
for solutions of the equation of motion. The scalar product (ϕ, ϕ)t is also time-independent, as one easily checks using the equation of motion (3.2):
for ∂V → ∞ and ϕ → 0 at infinity. In fact, the probability density satisfies a continuity equation analogous to that in non-relativistic Quantum Mechanics:
where
We may in fact collect
where
Although we have succeeded in satisfying the requirement (i), this is not the case as far as requirement (ii) is concerned. As a simple example shows, the probability density is not a positive semi-definite quantity. To see this, consider the superposition of two plane waves for two free particles of mass m:
with k · x = kμxμ, etc.
Let c and d be real. One then has