233 Other Interaction Terms in Perturbation Theory and Scaling Let us look at

233 other interaction terms in perturbation theory

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2.3.3 Other Interaction Terms in Perturbation Theory, and Scaling Let us look at what sort of effect we would have from the interaction terms we ignored. For example, we discussed a possible term g 4 φ i - φ i +1 a 4 g 4 ( x φ ) 4 (2.3.28) in the continuum limit. Let us consider a situation where the effect of this term is extremely small, so it can be treated in perturbation theory in g 4 . In such a case it is just a small perturbative correction to the Hamiltonain. Schematically, since x k , this correction looks like H I g 4 X k i δ ( k 1 + k 2 + k 3 + k 4 ) k 1 k 2 k 3 k 4 ( a - k 1 + a k 1 )( a - k 2 + a k 2 )( a - k 3 + a k 3 )( a - k 4 + a k 4 )(2.3.29) The integral over x gave a momentum conserving delta function, the x factors became k i , but most importantly, this is an interaction among 4 sound waves . If we included it, they wouldn’t just pass through each other anymore, but could scatter. Note that although it appears that it could create or destroy four sound waves out of nothing, that actually can’t happen, due to momentum and energy conservation. Notice that this term is of order (momentum) 4 . Another example of such a term would have been ( 2 x φ ) 2 k 4 φ 2 k (2.3.30) Since we are studying the long wavelength limit, λ → ∞ , we want k 0 (or more precisely ka 0), and so higher powers of the momenta are smaller. So by ignoring such terms we are not really leaving anything out – we are just making a long-distance approximation . 16
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2.4 Special Relativity and Anti-particles For our purposes, special relativity is tantamount to the statement that space and time have certain symmetries. 2.4.1 Translations The most obvious part of those symmetries is translation invariance – the fact that the laws of physics are the same everywhere, and at all times. Let us consider what this means in an extremely simple context, namely just the space of functions of various spatial variables x i labeling a bunch of points scattered in space. If a function f ( x 1 , x 2 , · · · , x n ) is translation invariant, then f = f ( x i - x j ) (2.4.1) Translation invariance is the reason that momentum space (and the Fourier Transform) is a useful idea. We have that f ( x + y ) = f ( x ) + y∂f ( x ) + · · · = e y∂ x f ( x ) (2.4.2) so we translate a function using spatial derivatives. But in Fourier space f ( x i ) = Z -∞ n Y i =1 dp i 2 π ! ˜ f ( p i ) e - ip i x i (2.4.3) which means that ∂x i f ( x i ) = ip i ˜ f ( p i ) (2.4.4) so momentum space linearizes the action of translations . Instead of having to compute a (spatial) derivative, in momentum space we can differentiate by simply multiplying by ip i . This exponentiates, so moving f ( x ) f ( x + y ) just requires multiplication by the phase e ipy . Translation invariance implies that f ( x i - x j ) = Z n Y i =1 dp i 2 π ! ˜ f ( p i ) e - ip i x i δ ( p 1 + p 2 + · · · + p n ) (2.4.5) In other words, translation invariance implies momentum conservation, and vice versa. Momentum and energy are conserved because the laws of physics are invariant under translations in space and time, respectively. We will explain this again in a more sophisticated way very soon. In general, the
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