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Lie Groups #7 - The Special Unitary Group SU(2)

In this video I introduce the special unitary group SU(2). I show how we can realise this as a matrix group and show the form of any matrix that will satisfy the general definition (this is easily derived but I skip the algebra here). I show how these 2x2 SU(2) matrices (that are parametrised by two complex parameters) can be understood as having four real dimensions, and derive the constraint that these coordinates must satisfy. I then go on to discuss how we should realise these group elements geometrically, where we discover that the group elements can be realised as lying on the unit sphere in four dimensions S^3 ⊂ R^4. In upcoming videos we will discuss this in much greater detail, when we realise that SU(2) elements naturally act on unit spinors, which can also be realised geometrically as points on the unit three sphere. Notes are on my GitHub! github.com/rorg314/WHYBmaths Everything I make will always be free, but if you like my videos and want to consider supporting the channel I am now accepting donations in DOGE and BTC (other cryptos on request) See my channel description for addresses.