Can we add a normalizer while adding a new field to an existing index. I tried to but it doesn't seem to be working

The index is already in elasticsearch, I'm trying to add a new field "my_field" with a custom analyzer but this doesn't work.
PUT my_index
{
"settings": {
"analysis": {
"normalizer": {
"my_normalizer": {
"type": "custom",
"filter": ["lowercase"]
}
}
}
},
"mappings": {
"default": {
"properties": {
"my_field": {
"type": "keyword",
"normalizer": "my_normalizer"
}
}
}
}
}

I think it's not possible to add a new analyzer "on the fly".
One solution, is to create a new index with your new analyzer and field. Then you can reindex
your old index into the new one.

https://www.elastic.co/guide/en/elasticsearch/reference/7.1/docs-reindex.html

bye,
Xavier

Okay. Doesn't it make sense to allow adding a normalizer while adding new fields? Is there any reason, why it is restricted?