Variables in Batch Apex

I'm reworking some batch Apex in my org and am having some issues accessing class level variables. An example of what I have is:

global class myBatchClass implements database.batchable<sObject> {
    global final map<string, sObject> objMap;

    global myBatchClass (){
         objMap = new map<string, sObject>();
         //iterator to populate map
    }

    global Database.QueryLocator start(Database.BatchableContext BC) {
        return Database.getQueryLocator('<some_query>');
    }

    global void execute(Database.BatchableContext BC, sObject[] scope){
        system.debug(logginglevel.WARN, '~~~~~~~~~ ' + objMap);
        processRecords(scope);
    }

    global void finish(Database.BatchableContext BC){}

    public static void processRecords(sObject[] scope){
        system.debug(logginglevel.WARN, '~~~~~~~~~ ' + objMap);
    }

}

When trying to save this class, I get errors about Variable does not exist: objMap in the processRecords method. If I comment out attempts to access objMap in that method, the code compiles. If I then execute the batch, I can see the debugs from the execute method.

So, what am I missing? Why can't processRecords() see the variables that were instantiated/populated by the constructor?