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Understanding Local Gauge Transformations and Field Particles

The text discusses the Lagrangian density for gauge particles, mass terms, scalar particles, Higgs mechanism, and interactions between fields. It delves into symmetries, vertex couplings, and spin characteristics. The concept of local gauge invariance is emphasized.

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Understanding Local Gauge Transformations and Field Particles

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  1. L=½ + ½ ½(2v)2 v2+2¼2+2  + ¼v4 ½  ½(2v)2 Explicitly expressed in real quantities  and v this is now an ordinary mass term!  “appears” as a scalar (spin=0) particle with a mass ½  “appears” as a massless scalar There is NO mass term!

  2. Of course we want even this Lagrangian to be invariant to LOCAL GAUGE TRANSFORMATIONS D=+igG Let’s not worry about the higher order symmetries…yet… free field for the gauge particle introduced Recall: F=G-G  *  12 + i22 again we define:  1 + i2

  3. Exactly the same potentialU as before! so, also as before: Note: v =0 with

  4. L= [+ v22] + [  ] + [ FF+ GG] -gvG 1 2 g2v2 2 1 2 -1 4 +{ g2 2 gG[-] + [2+2v+2]GG  2 + [2v3+v4+2v2 1 2 - (4+43v+62v2+4v3+4v2 + 222+2v22+v4 + 4 ) ]} L= [+ v22] + [  ] + [ FF+ GG] -gvG 1 2 g2v2 2 1 2 -1 4 g2 2 +{ gG[-] + [2+2v+2]GG  4  2 1 2 - [4v(3+2) + [ v4] + [4+222+ 4] and many interactions between  and  which includes a numerical constant v4 4

  5. The constants , v give the coupling strengths of each                  

  6. which we can interpret as: massless scalar  scalar field  with free Gauge field with mass=gv L = + + a whole bunch of 3-4 legged vertex couplings - gvG + But no MASSLESS scalar particle has ever been observed  is a ~massless spin-½ particle is a massless spin-1particle spinless,have plenty of mass! plus - gvG seems to describe G  • Is this an interaction? • A confused mass term? • G not independent? ( some QM oscillation between mixed states?) Higgs suggested:have not correctly identified the PHYSICALLY OBSERVABLE fundamental particles!

  7. Note: • Remember L isU(1)invariant • rotationally invariant in , (1, 2) space – • i.e. it can be equivalently expressed • under any gauge transformation in the complex plane or /=(cos + isin )(1 + i2) =(1cos-2sin ) + i(1sin+ 2cos) With no loss of generality we are free to pick the gaugea , for example, picking: /2  0 and/ becomes real!

  8. ring of possible ground states 2  1 equivalent to rotating the system by angle -  (x) (x) = 0

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