| Summary: | The amplitude T for ‘free-free’ processes such as bremsstrahlung or photoabsorption by an electron in the continuum in the presence of an external field, is usually written as the matrix element of the radiation operator taken between two continuum states. However, unlike the case when at least one of the states is bound, as in radiative transitions, electron capture or the photo-effect, this expression contains unphysical term, proportional to a delta function, and is not really the physical amplitude <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mrow><mi>p</mi><mi>h</mi><mi>y</mi><mi>s</mi></mrow></msub></mrow></semantics></math></inline-formula>. We first give an a <b>priori</b> definition of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mrow><mi>p</mi><mi>h</mi><mi>y</mi><mi>s</mi></mrow></msub></mrow></semantics></math></inline-formula> in terms of the scattering parts of the continuum functions, which does not have this delta function term and has an obvious interpretation in terms of time-ordered diagrams. We then show that when the formal amplitude <i>T</i> is modified by a long-distance cutoff, the modified form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mrow><mi>α</mi></mrow></msub></mrow></semantics></math></inline-formula> approaches <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>T</mi><mrow><mi>p</mi><mi>h</mi><mi>y</mi><mi>s</mi></mrow></msub></mrow></semantics></math></inline-formula> as the cutoff is removed. The modified form may be used as the basis for calculation and approximations without the need to introduce further cutoffs at a later stage.
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